Relative with Alzheimer's is in a very small MC facility, sharing a room with a stranger. She wants to go home, I want to move her.

I would switch her to a private room . I have seen bad behavior by other residents at Nursing Homes .

How long has she been at the memory care she's currently in? If it's only been a few weeks or months, she will need time to adjust to her new environment. So you could wait it out for a while and see if she improves at all with anti-anxiety meds. They should definitely be medicating her some to help with her being combative and anxious.

I'm always a believer in a person having their own space. Would it be possible for her to have her own room at the MC she's in now?
Don't make the mistake in thinking that a bigger facility means better quality care. It doesn't.

Regardless of what facility you put her in she is going to want to go home. She may not even be thinking of her most recent home. With dementia and wanting to "go home" they could mean a childhood home and a time in their life when they were young and happy. The begging to go home will be a given that will likely continue until her Alzheimer's becomes more advanced. There's nothing to be done about it.

Anti-anxiety medication can help her. They are even prescribing anti-depressants to people with Alzheimer's as well and they have been showing promising results in helping with combativeness, agitation, and anxiety. Talk to her doctor. Try medication before finding her a new place.

If you move her to another facility she will
Want to go home
Never shower (going to come back to that in a bit)
She will be terrified (even more so if you move her)
Be Unhappy.

No matter what you do, where she goes the scenario will not change.

I do have a problem with the "she will not shower"
I do hope that the staff is giving her a shower. If they are not helping her do the ADL's that she can no longer do then it is time to have a talk with administration as this is what the staff is supposed to do.
She can say "I don't want a shower" but the staff in a MC facility should know how to get a non compliant resident to comply.

Touching back on the terrified part. Is she on medication to help with the anxiety and the unhappiness she is experiencing?
Is this facility trying to engage her in activities?
If not then a talk with administration.
If talks don't work for either and a complaint to the Ombudsman does not work then you can think about moving her to another facility that WILL.

What medication has been tried? Any?
Because in my personal opinion she might be getting more caring and careful care where she is.
She is not going to change wanting to go home. That won't happen.

I would discuss possible trying a low dose anti depressant after a talk with the doc, or even low dose cannabis. Many are having good results with the latter.

I have an awful feeling that outdoors, which she is unlikely to access very much without staff having time, and a private room where she is alone is unlikely to help.

Only move her if she's able to pay for it completely out of her own funds. If she's on Medicaid, this may be why she's in a shared room. You can maybe find a better Medicaid-accepting facility but she'll still be in a shared room. Many facilities have waiting lists for Medicaid beds and the existing residents get first dibs on them.

Have you discussed her conditions with the admins? Maybe give them a chance to correct the situation first. Hint that you will register your unhappiness with their facility with their governing accreditation organization, or on Nextdoor.com, or both.

FYI, wanting to go home is called Sundowning and she's referring to her childhood home, not her most recent place of residence.

Is she on any meds for anxiety/agitation or depression? If not, why not? They often help a lot. People with dementia can't bring themselves out of those mental conditions because their brains are "broken".

She'll be unhappy no matter where you place her, most likely. But I'd move my loved one to an MC with a private room and caregivers who know how to bathe and deal with uncooperative residents, if it were me. Anti anxiety meds are something to look into as well. Ativan helped my mother quite a bit.

Good luck to you.