Mother (89) lives with us since Dad died 4 years ago. She tells everyone this is half her house though she pays no utilities. Any advice?

Now that you have accepted money from her for a down payment she likely doesn't have enough left for a good assisted living? I am guessing? You did know that she was demanding and controlling and you still chose this way forward? It sounds then as though this bed is somewhat made.
As to her having someone to stay there when you are gone, you can say whatever you like. You can tell her that it has nothing to do with HER but that you are hiring a housesitter to be certain all goes well at home while you are gone. PERIOD and exclamation point. This isn't her choice, it is yours. You will be gone for the weekend. You need to set some clear boundaries.
As to what she says about whose house it is, why in the world would you care? The deed indicates who the house belongs to. Whatever IT says is the truth, so why would it matter what Mom says? Let her enjoy it, tell her she did a wonderful thing. And you love her. But, hey, you are going away for the weekend. Be sure that your housesitter has sets of keys to get in and a letter from you in case Mom decides to lock him or her out.

I agree with AlvaDeer. Unless your mother’s name is on the deed to the house, she doesn’t own it even if she gave you enough money to pay for 99.9% of the home. Trust me, “everyone” could give a rat’s tukis less about who actually owns your home. I have to question why you would let yourself be goaded into uprooting your lives and leaving your home for someone you say has always been “demanding and controlling”.

Before I went on any trips or made any large, unnecessary purchases, I would open a savings account and make deposits into it to pay her back for any monies lent to you for the house.

So, is your question of "Any advice?" about what to do with her notion of being part owner and controlling you with that? Or what to do about wanting to extract yourself from the situation while still making sure your mom is cared for?

Not sure what your mom's cognitive abilities are, but as long as you are indebted to her she will always believe she has a portion of control and a say in things, even if it's just in her own mind. Without getting into the financial weeds, as was suggested below you could begin to "charge" her for her portion of the home owner's insurance, property tax, utilities, upkeep, etc. (on paper) and put it against the down payment "debt" (on paper) and just keep a running tally until that debt is back to $0. Let her know that you are doing this and let her know when it has reached $0.

If you're wondering how to get your life back...you won't be able to unless you insist she receive care from alternate sources so that you can live a normal life. AlvaDeer's suggestion to, "Be sure that your house sitter has sets of keys to get in and a letter from you in case Mom decides to lock him or her out" is necessary, as well as a call to the agency making sure your mom has no authority to "fire" them in your absence. Make sure she knows that if there are any shenanigans while you're away, the house will go up for sale and you and hubby will downsize to a 1-bedroom home (hint hint). Of course, this is all predicated on her ability to truly understand what's going on and being able to make rational decisions. Good luck!