sounds like the old opposed twin briggs. most didn't have a fuel solenoid so you may not have one. that carb seems to have problems with any small amount of junk in the carb will cause run problems. it could be a case were the fuel pump is not pumping fuel when it gets hot. had one on a intek last year in the shop that did that but it was the round plastic one above the valve cover and not mounted to the carb. most likely a fuel pump problem or something floating around inside of the fuel bowl.