I would be recording all those positions relative to that base position on the ground as you are, then retrospectively applying the shift to it’s known point once you have separately processed that.
This is normally done by via what’s colloquially known as PPP (AUSPOS for example does not use PPP techniques and is a network differential solution) but if you subsequently feed enough of your base data into one of the services it will provide you with a known point.
Emlids guide for this is here:
How to correct collected points with a new base position? - Support tips - Emlid Community Forum