How should I handle the car key situation?

I had this issue with my father. We talked it over, and he agreed to be assessed by an outside party. People usually believe paid consultants more than family. We found there were policemen willing to give assessments in their spare time for a small fee. Of course I stacked the deck by talking to the policeman ahead of time, but I do not think it was needed.
Soon after the assessment we removed the car when he was receptive. Eventually we gave it to my daughter and let my father be a hero, but I also would have sold it to a business who buys cars as I did with my FIL's car.
Your father would need to be present to sign the transfer, but that is only a one time project, it is better than replaying the battle again and again. Looking back on the before and after, I would have kicked in some money to help make it happen.
Playing up the positives, such as the money from the sale, helps. Going to lunch or some other positive activity afterwards is also good. Getting a good price is not important as getting rid of it quickly.
The battles fade when the car is no longer there.

Disable the car (put it on axle stands, take off the distributor cap if it's got one, e.g.). Then at the very earliest opportunity, e.g. when your mother next shows awareness of her dementia, agree its removal and sale and bank the money for her.

The trouble is, a dead key might stop her actually driving it onto the road but if you can get into a car and release its handbrake and steering lock you can still do a heck of a lot of damage. Seen it happen. And that's apart from the worry and anger she might experience when she can't make the key work.

Also: where does she want to go most regularly, and what could you set up as alternative transport for her?

My mom knew a woman who killed a kid that was riding his bike. It does happen! Comes to a point it is dangerous to drive and the family has to stop them from doing so. If the person doesn’t have a family who stops them?

I was lucky I guess, my mom and dad did not fight me about driving and we gave the car to my brother.

My friend disconnected something on her mom’s car, thinking that it would solve the issue. Her mom called AAA out and they fixed it. She started driving her car again, so I doubt the dead fob would work. She will get it replaced with new one.

Notify her car insurance company about her diagnosis. It will be up to them whether or not they are willing to insure a driver with dementia. Without coverage, you may have an easier time with your mother about selling the car. On days she understands, you can blame the car insurance company. On days she doesn't, no amount of explanation will satisfy a person with dementia.

If the car is in good shape, the sooner you sell it the more money you'll get for it, which will help pay for care.

Whatever you can do that will satisfy her at that time is ok to me. 😉

You mentioned she is paying for this car. If it is not safe for her (or others around her) to be driving, why keep the car at all?

The only issue that I can see with giving her the dead fob is that one of her helpful friends might look into replacing the battery for her.

Whatever works, get her off the road. We kept my mom's car when she got lost a couple of times, but asked her to please not drive. She had stopped at my son's house to ask how to get to the bank. He drove her from his house. Then she didn't know how to get back home from his house. All this within 10 city blocks. Another time she found herself way west of her house and stopped to call me for directions. My son and I went to get her. She insisted that she would drive when she wanted, but just once got no further than the next street and realized she had no idea where she was going, so returned and never tried it again. We never had to hide the keys. Her car was used to take her and her dog places. If your mom has early dementia she might be convinced to not drive now and be afraid to try later. It depends on how willing she is to admit to her dementia.

It would be better to remove the car. My aunt said if the car was there she would drive it. Of course she might just start asking you were is the car?

My family member is a retired fireman. He has been to many accidents involving old people, most of whom were 100% in the wrong. Why? Because someone said old person, it's their fault. Yup, dementia, it will be your mom's fault, no matter what, because she won't be able to prove otherwise. Is it worth it? Not for our family. I got rid of mom's car.
Don't know if you remember or even heard of what happened a number of years ago where an older person with dementia hit some people, didn't even realize he did it. The poor man...and the people he hit.
If I were you, have her take a driving test or lesson, or have the DMV or doctor make that decision for you, so it's not your fault that she cannot drive. That would be heartbreaking too. Dementia sometimes turns into a tape loop, the same scenario or thought will keep playing in their minds. You cannot feel guilty about that. Best thing, go to her next doctor's appointment and ask him straight up with mom beside you, is mom still capable of driving safely?

you do whatever you need to , to keep her safe. My dad once wandered out to the garage 20+ times in one day ( hubs counted) to ask if that was his car out there, and where the keys? It was non stop repetition.. We finally told him "he lost the keys" and it would take awhile for the new ones to arrive." It is a long hard slog.. but eventually they forget.. If we are lucky! My FIL left my obviously impaired MIL drive until she ..A hit a kid on a bike ( luckily not hurt but very angry) and B rear ended a carload of day workers who called the cops! Even then he was willing to take her to renew her license. Thank God the DMV sent a letter saying she needed a full driving test to renew.. he never took her!