[DG Boxster]

The side windows drop immediately when I open the top latch. That's why I don't think it is the microswitch up at the latch.

Pushing the dash switch multiple times, or pushing harder, makes no difference. If I hold the dash switch continuously, the top will eventually start moving. Once it starts I can let go of the dash switch(motor stops) and press again (motor starts) and it all works fine. That's why I don't think it is the dash switch.

I think if I start the car, wait 5 minutes, then pop the latch and press the dash switch, it will probably work fine. I need to test more.

It feels like the body computer (or whatever controls the top) is rebooting or something after I turn the car on.

The battery is an interesting idea - I haven't had it tested. Starts the car fine. This problem will happen even with the engine running (and alternator presumably charging).