A steel flat bar across the compartment door with 2 padlocks attached to hasps on either side will probably deter the wandering thief. He'll just choose another trailer for what he wants. It occurs to me that you could move the items you typically store in the compartments to the inside. Leaving a compartment bare is a good indicator to the crack head that you have removed any easily accessible items.
If a thief is determined there is no stopping them. FWIW, I contracted in a local petro facility. Of course, the refinery is fenced and the usual security is in place. Overnight, thieves came in and removed 1/4 mile of large copper electrical cable. It was powered while they removed it and fastened to a walkway leading down to the river. A lot of work to accomplish just for copper wire but compulsion to steal overrides most reasoning, I guess.
Good luck with the security upgrades. Keep it visible and remove valuables would be my suggestion.