So what is the problem again..?
It starts Ok, after a few minutes it starts to break up, but clears up with the choke 1/2 on, right? That indicates fuel delivery problem,,,either dirt inside the carb circuits, faulty fuel filter, bad fuel line, bad high speed adj, bad pump (but I don't think you have a pump ( for some reason I can't see pictures on GB's...they don't pop up for me), or crap in the fuel tank. It also indicates a sticky main jet or float adjustment problem...i. e. the fuel bowl runs out of gas, or high speed adjustment screw (if it has one) needs to be backed out. Could be electrical like bad plug or coil when it gets hot, but unlikely since 1/2 choke seems to help the engine run. I've had this exact same problem with these tecumsheits.....They're Ok but not like those old trusty Kohler singles.
You put this engine on, did it include a new carb or is that the old carb and old tank..? Anyway, you expecting snow soon...??