Well, the factory carb can work just fine. Staton-inc sells a kit that helps you adapt a throttle to the GXH50 carb: Throttle cable Assembly for Honda GXH50 QXA
The Honda carb has a slightly small jet in it compared to the HS/Titan carb. If you run the engine bone stock it should be fine. But once you start doing stuff like freeing up the exhaust and intake, the engine may start to run lean. Happened to my brother's GXH50. Simply drilling out the jet or swapping it for a larger one will take care of that.
If you really want another carb, the HS or Titan carb will work. But seeing as how you start with a perfectly fine carb, I think the Staton adapter kit would be ideal. If you do find yourself having fuel delivery problems, then a carb with a pump diaphragm might be what you need. I can't find a source with one at the moment tho.