Gas cap probably isn't your issue. If you had problems, then opened the gas cap and then your problem was gone, it would be suspect.. I've seen it before where they don't vent or don't hold vacuum correctly on some designs and it can cause issues.
Couple things with an electric pump:
1. If fuel is boiling in the line before the mech pump, then it can be tough for the mech to get a prime and deliver fuel.
2. If the fuel is boiling after the pump and not filling the bowls, the increased and constant pressure from an electric can reduce fuel evaporation. Turning into a gas is a function of pressure and temperature. Higher pressure means you need more temperature before it boils (this is why water boils at lower temperatures at high altitude).
Without a return style regulator, it can be tough to build enough pressure to prevent boiling, but the steady nature of the pressure helps. There are return less style regulators, but they seem to be of questionable performance - they can help build a little more pressure just prior to the carb though.
If everything seemed to run well except WOT once it was good and warm, then I'd begin to suspect something much much simpler: float levels. If the fuel level in the bowl is too low, it will be tougher to start warm (even less gas left after the engine cools), it will stumble on moderate to full throttle, and can contribute to other issues.
Proper float level is dependent on the model of the carb. It's also very easy to adjust, but shouldn't be done blindly. Measure properly and document the changes. Also have a few good gaskets on hand in case you break one tearing into it 😉