To help with fluctuating temps, you could cover the sides of the tank with 3/4 inch insulation board (comes in pink or blue). It should help boost temps and stabilize things as well.

I would minimize handling as much as you can. You could do the pillow case method (maybe someone could find the sticky for you) to remove some of the shed but I would think handling and running water and such would be quite a load of stress since its immune system is working overtime anyway.

I have limited experience with RI, but the resuce that I took in who was near death, took at least a a week and a half to show significant improvement on Baytril injections. I assume it will vary depending on the sevarity of the infection.