You can pump through a mechanical pump, no problem- the only restriction in a mechanical pump is to the direction of flow. The issue is the style of electric pump you use- whether it will allow the mechanical pump to "draw" through the electric pump when it's not being used. Diaphragm pumps should work, roller/rotary pumps may not.
For this, you don't need a big expensive pump, either; since it's just being used to "prime" the engine, not "feed" it. I have used the little cheapie $30 dollar electric pumps from the Ag supply store, just make sure the one you get has adequately sized ports- I have seen small 1/4" sized ones that are a no-go.
But before you do all this, confirm (as Dana67Dart said) that your fuel bowls are actually drying out. Early carbs with direct-to-atmosphere bowl vents are much more prone to this than later carbs with mechanical or electronically controlled vents, or sealed systems (think charcoal cannisters).