I just recently gave Bella a bath in our tub for the same reasons (too cold outside). My experience leads to giving the following advice: a detachable shower nozzle will DEFINITELY help!, if posible secure him to something so that he doesn't try to escape the tub while washing (Bella ransacked my house, ate all the cat's food and generally wreaked havoc in a matter of minutes becuase I couldn't hang on to her), and if Teddy get's the zoomies, watch out!!!
I am battling a skin problem of some kind (just lots of itching). I tried bathing her to make sure that it wasn't something on her skin, but that didn't work. So, the vet has given us some special shampoo to try. Looks like I get to do it again tonight!!! Good luck with Teddy. Another option would be a petstore that has the rental tubs that you can wash Teddy in yourself.