If you use the trailer to haul anything that is sold or in competion that has any sort of payout then you are commercial once you go over 10,000 per USDOT. SO you would need the CDL class A. If this is all just for fun recreation then the Non-CDL Class A would be fine.
If you are over the magic 10k number IE any dually or a 3/4 pulling almost any trailer then you must do the DOT number, Physical, log books, **** test etc IF you are using the vehicle for any sort of commercial operation.
This is per USDOT there are many people flying under the radar and have no problem but they are finding there is a lot of reveunue to be made by all the fines they can collect on being in the 10,000 to 26,000 pound range vehicles. Big issue is the burden of proof that you are not commercial falls back on driver and you would have to defend the tickets.
Couple examples that USDOT gave me that are extreme but they were targeting
craft fairs where grandma and grandpa have an RV and haul a trailer with their crafts they ARE commercial and if over 10,000 in combo then they must have full USDOT compliance.
Race car hauler if the races have any sort of payout (whehter you earn anything or not) then you must comply.
Welder that has welder loaded on a 3/4 truck and is overloaded to 11,000 pounds is in major trouble he is overwieght and must unload before going any further (there is like 10% you can be overloaded and receive ticket and continue) plus must comply with USDOT.