If its on private property which you have permission to trap or adjacent to it, simply approach the USDA trapper yourself. He will very likely pull his gear and let you have at it. He doesn't get paid by the headcount, so you aren't costing him money. Most of their work is for preventing roads from being washed out, so it doesn't much matter to him who kills the beavers as long as they get killed and the road doesn't wash out.

He is getting paid to trap the beaver by the hour, so of course he won't mind if you come in and trap them for free. What I am talking about is USDA subsidized competition for the contract to trap them in the first place. Private trappers can prevent road from washing out as well as USDA.

Its not just beaver.

That's where it gets problematic. If its a DOT contract which covers the entire State, the problem for DOT is finding a private outfit that has the capacity to cover the entire State. Otherwise they would have to piecemeal service agreements with several trappers from all over the State.
That would be a royal PITA for DOT. One significant problem would be that all of the different trappers would operate independently of each other which would make DOT ultimately accountable for each of them.
Let's say one out of those many individual trappers doesn't work out and DOT fires them. Is there another beaver trapper in that specific area able to replace that guy? Maybe, maybe not.
DOT prefers things to be simple. They don't generally know much at all about beaver behavior or trapping. They know that USDA does and hires people who do. If things should go south with a USDA trapper, USDA is held accountable. DOT just has to make one phone call to USDA to fix it. USDA can send in a trapper from a nearby area to cover until a replacement can be hired.