The lazzaret(te) (what Jack White calls the 'lazzareto') is essentially an ouebliette. It's not meant for anything heavy or large. Its original purpose was for access to:
1. steering gear
2. stern-anchor rode.
I wouldn't put anything else in there, lest it upset the trim of the boat. The propane bottles are fine, provided they drain ONLY overboard with no leaks into the bilge area and provided that their drains exit the locker above the waterline given 'usual' (not merely 'optimum'' trim conditions. Ditto for the beer cooler, for different reasons but with the same aim in mind.