How should I handle the car key situation?

When getting a dangerous driver off the road, all’s fair.

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.

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.

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.

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.

I think you might unintentionally be straying into some ethical hot water, here.

Your mother is paying for the car (is that some kind of finance deal? Any way to wind it up or transfer it to yourself?). It is her car. Is her licence valid? Her insurance?

I'm not saying woo-hoo let her drive if she wants to; but you can't keep her property, allow her to carry on paying for it, and prevent her from benefiting from it all at the same time.

If I were you - in fact, this is more or less what I ended up doing, because I couldn't afford to insure both cars and mother was heartbroken at the thought of losing hers; so was I but we all have to make these little sacrifices humph - I should get rid of your car and buy hers from her. If you haven't already, it is also time to notify her insurers and your local DMV people.

From there, it is reasonable to ask her to pay a proportion of the running and/or finance costs equivalent to how much use she gets from the car. So, say half of your mileage involves taking her to appointments (that'd be a lot of appointments! - but you see what I mean), then she pays half.

Has she given you power of attorney, by the way?

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?

Why is it so hard to take away a key from someone who shouldn't be driving? You are the grown up now, they are like children. Just take the key and tell them you'll drive them where they want to go.

Her doctor didn’t tell her “no”? If they are aware and have diagnosed her, especially if she is on medication for dementia they should be your ally. It’s usually much easier if they tell her; somehow it’s easier when it comes from the doctor. Another way might be to call Motor Vehicle, and tell them what medication she takes for dementia, which is what my daughter did for her father. With the letter they send canceling the license you might inquire from the car company if they can give her a buy back or cancel her lease. Let her keep her dignity because she might feel that’s all she has left. I found everything is easier if you talk about things you’re going to be doing with your loved one, and make lists of things you want to do “with” them. It’s a lonely scary world when they start to lose the power they had to control it.

I worked on a Memory Care Unit . The patients could see the visitor parking lot from the dayroom windows. Several patients thought they "saw" their vehicle parked there. Some would demand to have their car key as they "knew" that was their car they saw. Took a lot of distractions and "fiblets" to redirect attention.
Having a car represents freedom and independence. One thing you could do is remove car from sight and say it is being repaired. Taking out battery is another way to prevent driving, but best that car is out of view.