You might have a master key. I think that for two reasons.
1- You say the key works in the handle but not the deadbolt. That is exactly what a dealer master key will do. A regular key for a camper will work both the handle lock and the dead bolt. I used to work for a RV dealer and had a master key albeit it was red vs green.
2- Looking at the pic of your key, it looks to me like the part number starts with MK which may very well stand for Master Key.
Have you actually tried a code 310 key? If the tag inside the lock says 310, a 310 key will work unless the lock cylinders were replaced. If you cannot find a key that will work, you can replace the lock assembly. Most are pretty standard among travel trailers and fifth wheels. IIRC a replacement lock assembly costs around $25 to $35 dollars, comes with at least two keys and can be replaced with a screw driver in about 5 minutes.
FAA A&P mechanic, ex-RV dealer parts dept
Retired B-52 Gunner/Flight Examiner, B-52 Crew Chief
2016 Jayco Eagle HT 29.5 FBDS w/ 15,000 BTU AC, 50 amp service, upgraded counter top, thermal pane windows, elec. stab jacks, outside grill, backup camera and MORryde pin box.
2011 GMC 2500HD DURAMAX/ ALLISON