It seems to me as if some bodyweight strength training exercises might not trigger any of these problems. I would suggest very small sets of comparatively high-load exercises, e.g. work your way up to one-legged squats, one-armed pushups, and chinups if you have a suitable thing to hang from or are willing to get a chinup bar (I sometimes do chinups on the metro). If you are interested I can give you some details on how to work your way up to these exercises, since many people are not initially strong enough to do them (I sure wasn't!).

Sweat: Short sets don't give you much of an opportunity to overheat or sweat. Also, with bodyweight exercises, you can do them at home and take a cool shower/bath immediately afterwards. (By the way, do you hate sweating, or do you hate being sweaty? I am assuming the latter for now.) You could even do the pushups in a cool bath to get some of the advantages of swimming.

Environmental Issues: You can do this at home and indoors.

Cost: Only the time investment, plus (optionally) the cost of a chinup bar.

Boring: This type of exercise does not take very long, so you won't have much time to be bored. Not very long means 5-10 minutes total, a few times a week. You don't even have to do it all in one session, you can take a minute at a time through the day.

As a bonus, strength training can make other sorts of physical activity less unpleasant, since you will be operating at much less than capacity.