HI, my Mom has Alzheimer's Disease and I have P.O,A, she was going to stay with my Cousin until my wife and found a bigger place got moved and settled went to pick up my Mom and my cousin said she didn't want to go with me. But right now you could convince my Mom of anything with little trouble
I would question why your cousin would just tell you something that a dementia patient told them as if that's the final word. If she's caring for your mom, certainly, she knows that mom can't be relied upon to make major decisions anymore. A more appropriate explanation to you might have gone something like this, "Cousin, your mother is doing fine here. When I told her you were coming soon to pick her up to move to the new house, she said she rather stay here. But, we know that she may change her mind at any moment. Why not let's pick it back up tomorrow and see what she says then? I have no issue if she stays with me, but that's your call. I'll have her ready when you say and we'll figure out a way to prepare her as best we can." That's what I would have done.
But, I'd consider if it's a good arrangement and make lemonade with it.
Have a plan in place for that time, when you & your wife will need to be in agreement on things like how you're going to manage 24/7/365 supervision of mom, transferring her, bathing her, changing her, dressing her, keeping her out of trouble and engaged in appropriate activities. The care proposition of an advancing dementia patient is not like nursing anyone with a broken leg. Think very seriously about what is to come and if you want to take this on as a new full time + job with no relief.
For now, if everyone's agreeable, let mom stay with your cousin. When it gets too much for your cousin or mom changes her mind, then take her to live with you.