My new share-o-plane has a PilotAware fitted on a cigarette lighter adaptor.
Saves messing with a usb battery like my previous rental setup, but being switched on with the avionics master means that it isn’t ready for use for a couple of minutes after startup.
I’ve got quite used to the engine start and off blocks logging, but this requires starting the flight before the engine is started.
Is there any way to start a flight using my iPad’s location services and then switch to the PilotAware once that has booted?
It should have finished booting by the time runup checks are completed...

I suspect it is not perfectly clear - my suggestion solely points to the GPS source fallback to make sure the tracking works. If Skydemon would allow a traffic source not also being the GPS provider, I personally would stop using it that minute. GPS position and traffic must come from one origin, otherwise we violate everything which made flying that safe. On fallback to internal GPS you loose all traffic addendum to your eyeballs, my 100percent opinion, full stop.

I’m afraid I don’t follow your logic...
If I understand the system rightly an ADS-B contact is providing its own GPS location that has nothing to do with your GPS receiver and a bearingless contact has zero impact on your location, although if I understand Tim’s comment above correctly the gps altitude is used.

As stated the Stratus can be built without GPS and apparently other providers can use separate location and traffic streams so it would appear to be technically possible.