How do you deal with a doctor who is not the least bit interested in helping mom get into a memory care facility which she definitely needs?

My mother didn't need a doctor's recommendation to get into her MC. The MC sent out their own nurse who evaluated her. She did the "count backward from 100 by sevens" test (passed), "Who is the president?" test ("The man with the pig eyes" Close enough-- passed), and just when I was expecting her to be rejected, my dear mama said to the nurse, "Have you met my [imaginary] husband? He's on the roof helping NASA with a satellite that crashed."

They welcomed her with open arms. :-)

If it really is as simple as one letter from Dr which you cannot get then use a lawyer to write a request for the letter - yes I know it costs (in the UK it would be about £150 which I guess is around $200) but not many Drs would refuse that request.
I agree with all those who have said look for another facilitiy, wandering is a definite need for in, when considering memory care - anywhere that will not take her on the grounds she wanders (which is debatable with the circumstance you give) is not one I would consider likely to fulfil other needs either, it seems to represent a lack of care.

Get another doctor

We had difficulty with a doctor at the time MIL moved into a NH. We did it anyway, then wanted to leave that GP and use the regular who visited the NH. Original doctor was very unpleasant and refused to hand over case notes. Other family members had trouble with him too, and we were told that friends were also in the process of moving to another GP. It seemed that his business wasn’t going well (not surprised), and he thought (correctly) that MIL going to NH would lose him another paying customer. It did provide a reason for the problem. The new doctor just took it from scratch, without her notes.

Change doctors! If she’s not seeing a Geriatric Dr., and a Geriatric Psychiatrist she needs to be ASAP! If she can afford a caregiver in the meantime during the transition please hire one. It’s just so dangerous to be home alone at this point.

At the hospital it is the social worker who will help you with placement. There must be diagnosis first that Mom isn't competent or safe alone. Social Worker can help you get guardianship or conservatorship if you are not already POA. Do not allow her to come home, because from there they will place it all on YOU to find a facility. I assume there has been no neuro psych consult and no diagnosis yet, and that has to come first. Good luck!

Has she had a needs assessment done by the facility?

I am confused by your post, because where I live, MC is the very place you would place a wanderer.

If your mom can live alone I would question her need as her doctor as well. Typically the person needs to be cared for 24/7 before a doctor will say that they need a facility.

Sometimes we just have to stop propping them up so that they fail on their own before we can get the help needed. It's a terrible system but, that's how it works.

She needs to be diagnosed as having a neurocognitive impairment or dementia in order to get placement at a memory care facility. These are locked units so the wandering really shouldn't be a issue, lots of people with dementia wander a bit.

A good memory care facility will be able to handle wanderers. This is not an unusual condition for people with dementia. Get a new doctor who specializes in elder care. Check out the facility carefully. They should be able to tell you how they have handled the pandemic. Ask about their protocols and whether they are able to quarantine people who are contagious. My mother's memory care facility reports number of positive cases for staff and residents and also if there was a death (there was 1). Also be aware that during the pandemic you may not be able to visit as much as you want. My mother's facility has to lock down the facility for 2 weeks (no visitors) whenever staff or residents test positive, which has been happening more often in this second phase of the pandemic.

Tell them your mom is danger to herself that she not only can take care of herself but she is now having hallucinations and needs to be in memory care.

If that doesn't work, get a new Dr.

She could have a UTI or just be dehydrated, which causes hulucinatiins.

But, those memory places are not great to live, especially at this time with Covid.

Your mom would be happier living in her own home where she is familiar.

It really doesn't matter if she doesn't remember if she ate or not.

She'll eat when she gets hungry.

My 96 yr old Dad eats a little every 2-3 hrs.

You could install cameras in her home and tell her to call you instead of walking over.

She may have just had a dream or saw shadows.

You should deffiently talk to her about staying in her own home or moving to a facility.

Those facilities are not what they make make them out to be.