The fuel pump is on the other side of the engine from the oil pump and driven by an internal diaphragm and a couple of "direction" valves which operate off a tube connected to one of the combustion chambers. It uses the different pressures on intake and exhaust strokes to move the diaphragm in and out thus pumping fuel. No oil here to amount to any thing. If you had a ruptured diaphragm you'd have high speed fuel issues....lack thereof.
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First thing to do is to premix your gas with 1pt oil to 6 gallons of gas. Flush your fuel line to get the oil into the engine.....I'd pull the carb drain plugs and use the primer bulb to force fuel through the system till you got "mix" into the carb. Then I'd pull the plugs and with Sea Foam Aerosol (auto parts stores carry it), a fuel additive, set the throttle to wide open. Spray generously into the holes while cranking for a bit...15 seconds. Sea foam is light mineral oil, naptha, and alcohol.
Reinstall and reconnect the plugs. Go to the carbs and spray into the venturis with the throttle wide open while cranking for a bit....15 seconds. It may light off, if so ok. It can run on the spray, or it will run on gas from the tank. That should get your engine protected while you figure out what's wrong.
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Follow the fuel line into the engine and you should come to a Y. This is where the oil is injected. The oil pump is between the Y and the tank. First I'd inspect for something wrong. Pinch off the oil input line to the oil pump.
Run the engine and look for something out of the ordinary. Report back.
