Oh.
I'd just like to echo something a PP said.
You will not catch a cold from your body getting a bit cold, it is a virus.
I normally bath my ds's every other night, or if I'm having a bad week, it might be every few days, but there is no reason what-so-ever to not at the very least give him a quick wipe down with a flannel. You don't even have to get him completely naked, do the top half first and then the bottom half.
BUT I feel this is only acceptable if they are bathing regularly.
Life is a struggle and tbh, I suggest you get used to it OP because it may get even harder as he gets older.
You have to do what needs to be done.
I live in a first floor flat and remember a while ago, going to the supermarket with my double pushchair and 2 ds's.
I don't drive and can only get a certain amount home on the pushchair and if I need more than I can transport home, I have to make 2 trips to the shop.
Then get all the shopping, 2 children (although easier now they can both manage the stairs) and a double pushchair up the stairs.
Saying in one of your posts that you try to play with your ds and that he has more toys than you can shake a stick at is irrelevant, he doesn't need toys, he needs to be clean and cared for.
If you feel you can't wash him in the bath, can't you put some water in the kitchen sink and do it there?
I will also add, that no matter what my dp said, if I felt we needed a heater to keep my child warm, I'd go and buy a bloody heater.