The mechanical pump has a set of valves in it. The valve that allows fuel into the pump should act as a check valve to keep fuel from draining back into the tank. However, I have experienced similar issues to yours on my old 63 and my current Lightweight. Both had original AC pumps. Both would sit for long periods of time between starts. I believe that the valves in the pump never seal perfectly and allow fuel to drain back slowly.
My 63 had a starter circuit separate from the ignition circuit, so I would simply crank the engine until oil pressure built, then turn on the ignition. For my SIII Lightweight, I found a simple solution. I installed an ignition cutoff switch that cuts power to the ignition and fuel shutoff solenoid at the carb. If it sits for a while, I simply shut off the ignition cutoff switch & crank the engine until the oil pressure light goes off, then switch on the ignition and it fires immediately. The added benefit is that oil pressure is built before firing the engine.
I did this because I didn't want to install an electric fuel pump.