I actually have a really similar setup. I do all my fermentation in the guest bath tub. I built a shelf that rests on the sides of the bathtub so that I can have stuff sitting on it and still use the tub for cleaning, rinsing, etc.
I want to build a ferm chamber to put on top of the shelf in the bath tub. I have a small mini-fridge and I'm considering using that, but I still need to be able to move the chamber to the laundry room for the 2-3 times per year that we actually have guests stay with us. I'm worried that a mini-fridge ferm chamber will weigh too much to move.
I also don't know if the SWMBO would be keen on me running electrical from the sink, across the vanity, all the way to the tub.
I also don't know if I might be risking electric shock somehow with this setup. The tub is never full of water, but I'm thinking that running electrical along the bathroom floor would have some risks.
The easier route is probably just trying to use the tub to do the ice bottle method, but I would need to somehow section off the tub so that I don't have the entire thing filled with so much water. I would have to section off the tub in some sort of non-permanent manner.
Let me know if you figure something out.