I don't think you need a specialized grey water pump; any small, submersible sump pump should do (smaller the better).
A submersible pump? Would you put it in a bucket or something, and maybe get a float switch?

Now, the apartment is on the second floor. Assuming the pipe runs straight down at least a couple feet, one could possibly stuff one of those check-valve-ended quick-siphon hoses down the pipe low enough to simply start a siphon.

I think gravity feed is the way to go. If I get a pump like that I can never brush my teeth or shave in the sink, which I may want to do, having only one each of wife and bathroom. It's true that the sink will be inconveniently high, but at least I'm tall.