Likewise with the M101A2 trailers, all three of mine have M101 beds on frames marked M116A2, the M116 is the designation of the bare chassis and depending on what was put on top the designation would change.
Texas law likes to use the numbers found on a vehicle's frame (not the body) for titling/registration purposes, so I would register the trailers with whatever info is on the frame (manuf, model, number, GVW etc...). the M103/M105 trailers ARE above the 4000lb GVW minimum requiring a title, and the 4500lb GVW requiring an annual inspection. Trailers under 4000lb GVW do not require a title but ALL trailers (regardless of GVW) are required in Texas to be registered and tagged, with road taxes collected according to GVW. My M105A3 will be registered Texas Former Military when I ever get around to doing it, so I can avoid the inspection and pay a reduced amount of road tax (I really DO only use it for shows and parades anyway...)