Only issue would be making sure to never power up the fridge before ventilating the space.
I'm not sure you gain a whole lot if you can't run it in the space (which is not likely set up for that). If you have to pick it up and walk it to an exterior location, doesn't matter much if it's in the truck or trailer.
We keep ours in the truck bed under a wood box and chained to the 5er hitch. Most would have no clue that it's even there and if they do, they have to be prepared to circumvent the lock...not impossible but a decent determent.
Keep in mind a trailer with no vehicle is easy pickings. Most storage space locks are identical and even if replaced, a small crowbar can open them in seconds.