How to deal with mean behavior and lying?

The family members are probably right that she has some kind of dementia. They can't diagnose Lewy Body or any other kind because they're not doctors.
Whether or not she gets diagnosed or if the cause for the abusive behavior is dementia-related really doesn't matter all that much.
No one has to tolerate and live in abuse. You have a little child and even though you are not abusive to her, you are allowing her to live in an abusive situation.
Right now, today you STOP and discontinue ALL caregiving to your husband's grandmother. You let him know and any other family members who need to know today that your caregiving is done.
Starting tomorrow, you start leaving the house with your daughter in the morning and do not return until your husband gets home from work. If you don't have family or a friend whose house you can go to during the day, go to the mall. Go to the library. Go to a woman's shelter. Does your child go to pre-school? That is going to make finding someone where to be during the day easier for you. If she's in school, go get yourself a job.
As of today, all caregiving services from you to the grandmother stop.
You do not feed her or prepare a meal for her. You do not help her with personal care like showering or brushing her hair. You don't help her with anything or do anything for her.
You stop cleaning her house, going to the store for her, and makig sure her errands are run.
Anything of value that belongs to you either has to be locked up or removed from her house.
You're not going to be charged with emotional abuse. Please don't worry about that.
The next time she throws something at you, grab her by the wrist and and get right in her face. Tell her that she will NOT throw things at you. Swear at her if you need to. I was an in-home caregiver for almost 25 years and had my share of abusive, violent seniors with and without dementia. A little profanity often works wonders on the abusive behavior of the elderly.
I had a client with dementia who was a biter and a spitter. She bit me hard one time and I slapped her. Not in the face, on her upper arm. Not hard enough to cause damage but hard enough to make a point. I got right in her face too.
She never bit or spit at me again and I worked for around two years for her after that.
It's time for you to grow a set and put an ultimatum to your husband. Either he moves out with you and your child or he chooses to remain with grandma and loses you both.
Stop letting your kid live in abuse. Mark me, grandma will turn on your child too at some point. What then?
An ounce of prevention is worth a pound of cure. Get your kid out of there.

Since your husband is not supportive of you and your daughter I would make plans to move out.
Talk with a lawyer first find out what the ramifications of that act might be. (Actually you should talk to your husband and if he does not know EXACTLY how you feel you need to tell him)
(can you support yourself and your daughter?)
(will you get child support? spousal support?)
(do you have a place to go? people to be with?)
(the important one..is this something you would do and follow through with?)
This is NOT the environment that you want your daughter exposed to. Your job as a mom is to protect your child.

Anyone can say anything but if there is no proof of what she is saying any investigation will go no further.

If your husband refuses to find in home care or assisted living for his Grandmother you need to take your daughter out of this abusive situation. Abuse is not just physical, this whole situation seems very chaotic.

you have a choice to remain or leave, your daughter doesn't. It is up to you to protect her.

Nothing makes me roll my eyes harder than someone claiming they "had" to move in with an elderly relative.

No, you didnt. You made a stupid choice. Undo it.

You are a negligently abusive parent to leave your child exposed to this behavior.

Hopefully if you dont leave you'll get reported to CPS and have your daughter removed from this abusive situation since you're unwilling to help your child.

Judging from your replies to people here… you aren’t going to do anything about this. You’ve made excuses for your husband and now you say the grandma isn’t all that mean to your daughter.

The one time you said you left, they begged you to come back. Why? You just walked right in to the situation again.

So… only advice is to get used to the abuse and accept you will be second to to grandma until she dies. Hopefully your daughter won’t be hurt, but that is just a risk you are willing to take. Oh well.

Lewy Body needs to be diagnosed as such. People suffering from it can become violent. You need to get grandma a good physical, neurological too. Everything you describe is Dementia but you need to know what type so correct Meds can be given.

Please be very careful with her around ur 4 yr old. This woman is paranoid.

OP, GMIL actually wants you and your daughter to leave. She wants her grandson all to herself. If you leave, you are doing what she wants. Make that very clear to DH and all his family. There is absolutely no reason for you to ‘feel guilty bc I've been caring for her for a while’. You can say quite truthfully that you aren’t happy and she isn’t happy, she wants you gone. Once you have gone, other plans can be put in place for her care.

Be very careful about moving back too easily. They may all have a huge interest in finding ways to get you back. Perhaps GMIL living elsewhere is the only condition that would help – and not just respite care, from which she could return.

I feel like I need to make another post because this really hits home for me.

your husband is a grandmas boy. That’s fine, nothing wrong with loving and having admiration for grandma! BUT you don’t come first, she does. Your his wife, you deserve to come first.

He is using you, she is using you and the family is using you. This is very damaging to your self esteem and most likely, unless a lot of work is done, your marriage will never be the same if you decide to stay.

Please ask yourself if you think you want this long term for you and your daughter

Those changes are not just PD.
PD is about diminishing motor skills, but not behaviour, unless she is getting dementia which is present in about 40% of Parkinsons.
So it is on your husband to get proper diagnosis and treatments. It is on him to tell granny enough is enough and you deserve better.

It doesn't matter if it is caused by dementia. It is not acceptable for her to abuse you. Plus this is not a healthy environment for a four-year old. She told you specifically that she is targeting you because she wants your hubby all to herself. She needs to be put in a nursing home. Or you and your family move out and she hires round the clock care for herself. Thats the only solution is getting you and your daughter out of this terrible situation. The rest of the family wont like it but too bad. They arent the ones being abused.

Sounds like you or your daughter will never be first. Your resentment towards your husband will only grow, long after his grandmother has passed away. No real man would tolerate this behavior for his wife. I’m going to repeat that, no real man would tolerate his wife being verbally and emotionally abused.

I have dealt with this my entire marriage (30 plus years) with my husband. I am telling you, that resentment does not go away. It scares and stays with you for life. Knowing what I know now, I would have left.

I guess you need to decide if you want to stay and be abused or leave.

Me personally, I would leave. Being abused on that level is emotionally and mentally damaging and lowers your self esteem.

It sounds like you do the work but really know nothing about grandma’s health. Does she have Lewy Body Dementia, or Parkinson’s, or both, or neither? Have you seen it in writing from the doctor? Who are ‘they’ that ‘tried to get a nurse’? ‘We both were told that she needed in home care’ – told by whom? Who are ‘the family that say we can’t’ move out?

Nothing should ‘make you feel guilty’. ‘Fear Obligation and Guilt’ are the biggest jerkers that keep carers in their place. Forget about being the perfect carer, in fact perfect anything.

It sounds as though your husband has no problems. He is treated like the golden boy, and thinks you must be exaggerating, should just ignore it etc. The rest of ‘the family’ are very happy with the way that things are just now. That leaves only one person who can change. You.

Change could range from moving out (as a holiday or a marriage statement) to just stopping all the work you do for GMIL. You don’t have to do it. Until you stop, no-one except you has any incentive to change. Perhaps you could tell us what you are prepared to change, apart from just wishing and hoping.

And that is no longer working is it? can her children help out as YOU are being abused? What does your hubs say about this? Well if CYS believes this stuff then you and hubs will have to move out,, and her other family will have to step up. Then you and hubs and daughter will be able to get on with your lives

Your husband is piece of garbage subjecting you and his daughter to this miserable, evil old hag.

Do you have family or friends you can move in with while you get a job and get on your feet?

Because the only way this stops is if the hag dies or you move out.

But really do you have any respect for your husband who clearly doesnt care that this is going on? I know I wouldnt.

Why does your family have to live with this woman? Could you save up and move out?