Why does my gas oven trip the GFCI when turned on after the clean cycle?
I'd suspect a small, varying current leakage to the metal frame.  Depending on how your kitchen is wired you might could replace all counter top outlets to GFCI protection, then remove the GFCI protection from the oven granted it is behind the oven amd not readily accessible.  However, doing so may allow a potential shock hazard.  

The best option is to have the oven checked out or replaced.

Also, I'd recommend putting the GFCI in the panel if possible or another readily accessible location if the one for the oven is not.