I don't know about another op picking up at Sunset House pier, it doesn't seen like something that would be done for several reasons. Aside from any competition thing, Cayman isn't a place where they run from pier to pier (most places don't even have them) to pick people up - the boat is waiting someplace convenient to where they plan to dive, and people are picked up by van or whatever. That's much more efficient there.
I would normally say it makes no sense to stay at Sunset House and dive with someone else, since IMO the only reason to stay there is the convenience of the whole arrangement. It's been longer since I stayed there but I was never too impressed by the dive op when I did. They certainly still have the same crappy slow boats they've had forever, actually worse since the Manta seems to be long gone. So if you're committed to staying there, it might be worth driving someplace else or getting picked up (by vehicle.) There are much better dive ops (and much nicer places to stay.)