How to override a stubborn father-in-law who won't agree to a care facility?

Write a care plan for caregiver to follow , for example: ADLs & keeping the area clean & maintained...or have Aide do laundry or go food shopping. Just keep track of receipts!

A cautionary word. If your LO has ALWAYS said he will NEVER go to a Nursing Home, there is a great risk he could die in the first 24 hours.
Grand FIL was struggling with staff trying to place him in restraints so he wouldn't leave, he had a massive coronary.
Grandfather died of a stroke the first night.
On answer mentioned Alcohol Abuse, you will not get him to accept care that keeps him from the booze, even though it IS killing him! Better bet might be an Assisted Living Apartment Complex, with additional services provided for support and safety. They might water down booze, or pour off a little bit into an old bottle throughout the day.
Remember studden Alcohol Withdrawal can be fatal!
Whenever my Mom was hospitalized, I called night staff RN and told he Mom was "a daily drinker who had sudden uncontrollable drops in BP 24-72 hours after previous admissions. Both parents in denial! Doctors need this information to treat her properly.
Hospital staff appreciate this. IF they site HIPPA, then start with, " In case you might happen to have a patient Name, Birthday, etc., Then their Doctors & staff need to know..."
They may not confirm LO is in hospital, but HIPPA does Not prevent them from receiving information!!!
Also, asking the front desk for nurses station for patient ___ , then ask if RN for patient ____ is available. Then Provide the critical information to them. Don't ask for Confirmation, unless you do so in person and are already over the HIPPA hurdle.

The caregiver is with them during the day so they have some coverage. You might have to bide your time to the next event then stick your toe in the door and negotiate for whatever you can get. If you start a routine for personal care for your mother on a regular basis, you might make headway on turning it over to the caregiver.

I went thru this with dad. He clearly was drinking too much, not eating, buying tons of stuff he didn’t need, and falling (he out his head through the wall) but refusing to go with EMS.
there is NOTHING you can do to force care on a parent. I contacted all the county agencies and was told “unless your father agrees to want the services, we can’t force him to accept the help.”
so we waited for the “big fall”.... and it was bad. He couldn’t stand or walk after wedging himself under the toilet tank between the sink and bowl for at least 8 hours. He ended up in the hospital, where we had plenty to tell the doctors. They medicated him, detoxed him, and deemed him “unable to live alone.” It was a blessing but also a curse. The problems didn’t go away, they just moved closer to me where I could keep an eye on them.
its still a journey ..... but it’s the best situation for now. Good luck!

What does their doctor say about it? Perhaps, their doctor could order some time in a facility for care, therapy, assistance, etc. If the doctor orders it, perhaps FIL would be more open. Maybe, he needs some therapy too, if he has poor balance, weak muscles, etc.

Regardless of why the FIL is not able or willing to take proper care of MIL, if I were the adult son, I suppose that's your DH, I'd seek legal advice from an Elder Law attorney to get the legal options. They can explain the process, what evidence is needed, costs, etc. so, you'll know what it required. And, if the situation is really dire, you could report it to APS for them to investigate.
Has the dayworker stressed concern about the MIL's care by FIL? If she's worried, I wouldn't just wait it out, because MIL having dementia prevents her from looking out for herself.

That may be exactly what you have to do. We often say here that it takes “an event” to put plans in motion. Where is your husband in all this? Can he talk to his father and maybe convince him to place his mom? If one of them should happen to be hospitalized, tell the social worker what you’ve told us. Did Dad fall while the caregiver was there? That needs to be reported. It may be that Dad won’t let the caregiver do anything.

Unless it's been determined that he's no longer competent, there's not a lot you can do other than wait for the " big fall".

Don't beat yourselves up. Elders have the right to make their own poor decisions and even the most loving family often cannot get the help needed put in place over the objections of an elderly parent.

If you think your MILs health is being neglected, call the local Area Agency on Aging and ask for a "needs assessment". Sometimes, not always, an elder will listen to an "authority".