That bit of smoke argues against it being a fuel supply issue, but that's still where I'd start. Basically you have two options, either it's the engine losing power (most likely fuel supply), or it's something in the hydraulic pumps, which is more complicated and more potentially expensive.
That's a simple inline injection pump that's been used for decades, and as reliable an engine you'll find in that size, but any engine can have fuel issues, especially if the fuel sits, like it uses only a few tanks a year or the fuel gets old. You could remove the fuel line to that transfer pump and blow back to the tank, see if fuel flows freely out of the hose if the tank is higher, and stick that hose in a bottle of fuel to see if it still stalls.
Otherwise, to look into the transmission hydraulics, I'd start at the filter or drain plug, but that's probably a mess. If you could find a manual, then pressure testing might be next, but that's not cheap or easy.