Is it common for the dementia patient to behave worse around the primary caregiver than other people?

I am actually glad that my Mom moved into Memory Care when she did. Completely different scenario as her husband passed away and her living with me didn't work out. She's pretty high functioning, but that seems to have helped her adjust to living in the Memory Care home, as opposed to waiting till things got even worse for her.
And yes, every change does seem to set her back a bit in the progression of her disease, but it progresses regardless, unfortunately that is the nature of it.

I think it is very common.

Yup, sure sounds like show-timing. Did you come across the observation that this takes a lot out of the person doing it, they can only do it for a certain length of time and that length gets shorter and shorter until they can't do it at all?

Your FIL's behavior around MIL is his "natural" state of dementia. The show-timing behavior is "unnatural" under the circumstances and takes a great deal of effort.

I'm so glad that you can be supportive of MIL. She is in a very challenging situation. (Been there, done that, didn't like it.) It is really great that you are doing some research on her behalf.

Keep this in mind if FIL does go to an appropriate care center. He will quickly stop show-timing for the staff (although he may still be able to do it for a doctor who comes infrequently.) This may mean that his behavior seems worse after he's been there a few weeks. Many in the family may conclude that going into care has made him worse. Probably not. Very likely they are seeing the state his mind has been in for some time but that he has been able to hide from them.

Mincemeat, thank you for your answer. I have been reading this web site, and many others, for over a year. This is the first time I ran across the words "show timing". Now that I know the word, I see many examples just like the one my family faces. It is now easier to understand why my mother-in-law is ready to move him from the home into a memory care facility. We will continue to help them both, and reassure her that we support her decision.

I appreciate your help.

Study up on "Show timing" They know when they must gather themselves together and behave. When they are on their own turf and around someone that they can get away with it, they let all the crazy hang out. Have empathy for MIL, it is wearing her down and frustrating her. Also, when dealing with somebody that does this, it is hard for others to believe you and be on your team. Please do not give her an attitude for her "attitude". It sounds to me that she is reaching out for your help.