Sorry about the late answer. Yes it can be done.
AABB-AABB intersections can be done by tracing a ray segment along each edge of each box against the other box (TLAS intersection testing).
Triangle-triangle intersection testing can be done by constructing 3 fake triangles orthogonal to each original triangle along each edge of the triangle (to catch the special case of the original triangles being coplanar) and then tracing a ray segment along each edge of each original triangle.