Symptoms point to system being overwhelmed by excess rain but if that were the case you would be complaining that toilet won't flush. You say toilet trap is full,what about shower,tub and basin traps? Unused basins and other drains must have water added to compensate for evaporation. You might try temporarily extending vent pipe higher and/or sealing window. It's possable that wax ring isn't sealing well and allowing gas to escape between toilet and floor flange. If flange isn't sealed I'd expect to see water seeping from beneath toilet. I don't have a 100% slam dunk answer.

Toilet has problems flushing but its a hit or miss. Sometimes it flushes fine. Other times it has trouble flushing or cannot flush; we wait a few hours and try again with success. Toilet is the only thing on the septic tank. Everything else is on graywater.

It is possible that wax seal isn't completely sealed, but I never see water seeping or pooling at base of toilet. Sometimes condensation happens.

The standard answer to this issue is to smoke test it.

That's probably the only way you will get a real answer to the problem.

Sounds like the main problem is rainwater overwhelming drain field. The fact gray water has been diverted from septic indicates this has been an ongoing problem. I see an aerobic system in your future.