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Apparently God Rod will /always/ give you the full benefit of the effect, even if you don't have a staff weapon or you have Priest as your sub-class. (As in your FAI will /always/ be added to your spell power with Curate/Priest spells. I only tested this with Shine Ray and Holy Spark, but, I'd imagine it'd go for all the other spells.)

I tested this numerous times to confirm.

Dark Water also doesn't seem to affect the FAI stat at all (at least via Black Bubble--the item made tiles weren't tested.)

Tested this with Holy Spark with a staff on my off-hand. This is happening. Whenever you equip a staff on your off-hand, however, with Priest is main- or sub-class, your damage will instead decrease. If you don't have a staff equipped at all, God Rod's effect still applies. Wow, what a doozy.

Without Staff:
--- ROUND 0 ---
Doctor Mavis Haemin's turn.
Jammer's turn.
--- ROUND 1 ---
Doctor Mavis Haemin's turn.
Doctor Mavis Haemin's turn.
Jammer takes 90 Light damage.
Jammer takes 90 Lightning damage.
The battle has ended.
Gained 1 Mura.
You consume your emergency food supply.

With Staff:
--- ROUND 0 ---
Doctor Mavis Haemin's turn.
Doctor Mavis Haemin's turn.
Jammer takes 79 Light damage.
Jammer takes 79 Lightning damage.
The battle has ended.

God Rod has been corrected.

Dark Water was also not working (I'm not sure how this hasn't gotten reported until now...)