My wife (also a doctor) actually started buying some bars for exactly that reason. Unfortunately they were too harsh. It's an unsolved problem, as far as I know.

Once there's enough demand, I'm sure manufacturers will follow the dollars.

I "solve" this problem by using soap but in smaller amounts. It doesn't need to form a lather.

I've been very happy with super-fatted soaps and soaps with added shea butter. My favorite so far has been Dudu Osun.

It's not common with refills for those bottles? (more often than not the refills are also in some kind of plastic - so not sure how much plastic it saves though)

Volume increases faster than surface area, so yes buying large refill containers will generally be better.