it ain't got a hand pump filter housing assembly on a L3400HST. It's got a cheap plastic housing with a filter inside of it no pump on it
it has a lift pump on the side of the block, no manual lever on it (unfortunately). Seldom failure point. Usually when the other techs suspected a failed lift pump, I'd ask them to check for fuel in the crankcase. If none, do a fuel flow test from tank to lift pump. Should flow on it's own (tank is mounted higher than the filter housing). If no flow, has a restriction in the line somewhere-or plugged filter or filter housing restriction. Just work your way back.
need fuel air compression to run. Which one does it lose within a second or two after starting initially? I am suspecting a shutoff solenoid or wiring or control issue but the issue was not described very well, so that's all we have to go on.