To a first approximation, I'd just calculate the convex hull of each triangle's start and end positions in the current timestep, and look for an intersection between the two hulls. True, this doesn't model high-speed rotation very accurately, and it's possible with the right circumstances that the triangles missed each other even if their hulls collide. However, this method might well be good enough for your purposes.
Another possibility is to look into using OBB trees. You can probably create an OBB tree that models the interior of your mechwarrior with enough detail, but is much faster than colliding with every triangle, not to mention probably easier to make a stable physics simulation out of.
Also, even if you stick with real per-triangle collision, you needn't use the same mesh for physics as for rendering. The physics mesh can be a very low-detail version and still get good-looking results.