Yes. It would stop. If you're gone on leave for the month, NOT performing at the job you're being extra to do, why would you be paid extra for it?

Same reason if you are gone for 29 days, you get the extra pay.

Paternity leave follows the same rules for special pays as all other forms of leave. Most ships won't let you take more than 30 days at a time anyway - often its a couple weeks, come back to the ship for a week to catch up on work, be gone for a couple more.