T3D has support for Physx and Bullet (as well as a custom rigidbody solver that dates back to the early 2000s) but there's no 'tick-a-box' option in the engine that enables BeamNG style physics. In fact, BeamNG is really just a soft body middleware solution that was ported to T3D for demonstration purposes and there was a lot of custom code involved to make that happen.
Godot is great for 2D games but I wouldn't use 2.1.x for any serious 3D project. 3.0 represents a big leap forward in terms of 3D capability but there's still many bugs to fix and workflow issues to iron out before I'd consider using it (we're talking months away from a truly production-ready release, in my opinion). Godot is on par with T3D in terms of physics which means you're going to need to dig into the source code of either engine to make what you want.