My grandmother demands that I bathe according to her schedule (despite the fact I own the house.)
Why?
Because she does not want to get up out of bed to DRY the bathroom. Yes, she DRIES the bathroom. She's convinced that if you don't religiously dry the tub, sink, ect., after using it, the walls will instantly become moldy and it'll cost a fortune to fix.
She refuses to listen to logic; every fixture, as well as the concrete board under the tiles, is waterproof and designed for moisture. (And its not like its going to be left with zero ventilation to dry out by itself.) And even if we did get mold, getting rid of it is a simple case of buying a bottle of bleach for $1.99. (My parents did that several times while I was growing up without issue.) Far from the "fortune" she makes it out to be.
How can I drive some common sense into her head?
However, you need to have your grandmother away from you asap. She may be in danger (apologize if I sound brutal as well). You perhaps did what you thought was right in the beginning but now it's seems to no longer be a good idea from reading your post. Nothing wrong with changing your mind or saying enough. Do what is needed for placement.
Sometimes for really annoying behaviors you may want to get clever. Can you change her clock so that it shows it's not time for her bath yet? Can you make up a "therapeutic fib" as to why she can't use that bath on that day (plumber is coming, the water is off, whatever works)? Is she capable of being the one to wipe it down? If so this would actually be good physical and mental exercise for her.
If you are being pressured into caring for her I think you're not morally or ethically obligated to do it -- especially if you're not getting paid. Not everyone is cut out for caregiving and there's nothing wrong with this. Just make sure you give your family enough time to find a replacement and, if you live in the same house, be prepared for the unhappiness and guilting. If you set up boundaries and stick with them your family will eventually get over themselves and respect you a lot more. May you gain wisdom and patience through it all.
Grandma has dementia, yes?
I know that you are a very bright guy; what have you read about dementia?
The thing is, my mom LOOKED like the same mom I had always know, but it wasn't her anymore. I SO clearly recall the day, in Bed Bath and Beyond when she was shopping for cortisone cream, because she had an itch. I handed her a tube of generic cortisone while she pulled down a tube of Gold Bond (which had no cortisone).
I explained to her about the ingredient list (this is the person who TAUGHT me how to read ingredients).
She said "but this one (the Gold Bond) SAYS "anti-itch cream".
I realized that my mom's brain was no longer firing on all cylinders and that arguing with her was pointless. I think I said "let's get this one mom, it's cheaper".
I cried all the way home that day.
RP, you need to get your head around that "explaining" and "reasoning" are not things you can do anymore with a dementia patient like grandma. You either go with the flow, lose your mind or get her into Memory Care, where the kind caregivers go home after 8 hours and get some rest before having to deal with craziness like this.
Have you looked up Teepa Snow and seen some of her videos? They might help.
"I also think that you are in over your head. This is not going to get better or easier."
If you are feeling so much stress over such a relatively minor thing then I doubt you are going to be able to handle it as things get really weird and dysfunctional. It's really okay to say you've had enough.
What time does she want to have her bath?
If it is really at an inconvenient time for you there are agencies that have "bath aides" and they will come out and give her a bath and help her dress.
It might be easier to do a shower rather than a bath.
I also think that you are in over your head. This is not going to get better or easier. I think it is time to look for Memory Care for her where she will be cared for in what might be a more kind and gentle way. (sorry if this is brutal for you) but it sounds like you are not cut out to be a caregiver at this point in your life. That may change but at this time grandma needs kind and gentle care.