I would rather have any hot air be removed from the aft cabin (and any battery fumes), especially in the summer down there. I'd install the quietest fan I could find. Many A/C fans are really 12 volt D/C fans with a wall converter and much cheaper and quieter than those sold as "boat fans".
Vent to a hose that goes outside the aft cabin to the lazarette, not the cockpit. The cockpit may fill with water one day and the water will have access to the boat's interior.
O2 Cool has a huge variety of fans that are affordable, almost completely silent, and last many years. You can even get them with USB power, or USB charging for its battery. I've run their 12" fans 24/7, and they lasted at least 5 years. It is very important to pick a fan that disassembles for cleaning. I couldn't believe how much dust the blades collected, over a short period of time.