akhdrider,
I am a licenced broker with thirteen years of residential real estate experience, including rental management.
If the missing or failed lockset is creating an unsafe or unsecure condition (impeding easy exit, not securing entry from unwanted persons, etc) then most states have regs dictating the owner remedy the issue. Do they always? Nope. The fact that you are renting a double-wide is indicative of the owner/landlord's commitment to quality and safety. In general, folks who own trailers as rental property are purely cash-flow oriented and will effect whatever actions are required to maximize it. (translation... they aren't in it to keep up the property and make you a happy, long-term renter. They are cheap &^%! who don't give a rat's patoot what's right, proper, or legal as long as they can get away with it.) Could you contact the regulating body in your state and get the owner to fix the issue under threat of legal action? Probably. But you would also likely find yourself in the crosshairs of a decidedly unhappy landlord who's just waiting for you to make a misstep.
In other words... Install the replacement lockset using threadlocker as other have recommended, then forget about it. You'll spend more time grousing about it than it will take to fix it, and nagging the landlord will only land you squarely on his bad side.
Brad