GFIC Breaker keeps tripping unless another breaker is off
Hunting those wascally cross-wired neutral wabbits

What you have here is a case of "crossed neutrals", where either:


Something on the lighting circuit is wired using a neutral from the GFCI circuit, or
The lighting circuit and the GFCI circuit have their neutrals connected (nutted) together directly.


The former requires a load on the lighting circuit though, so as long as there isn't a hidden hardwired load on it somewhere, I think we can rule it out.  The latter can happen on an unloaded GFCI due to the grounded neutral detector circuit kicking in, though, as it impresses a small signal on the hot and neutral lines that the N-G fault then disturbs.

I'd check every junction box the GFCI breakered circuit uses looking closely at how the neutrals are joined in them.  The lighting neutral needs to remain separate from the GFCI neutral in all places, or else you'll get a trip.