With any triangle, 8 copies can make an octahedron. With any acute triangle, 4 copies can make a tetrahedron.
Make a scalene octahedron, then construct scalene tetrahedra on each face. Here are samples with the 4-5-6, 6-12-13, 9-11-13, 11-12-15, and 11-13-16 triangles.
These are cases where we almost get a 16-triangle outer shell. None of them quite works, the outermost vertices don't actually coincide in these cases.
Is there an integer triangle that gives a perfect solution?