3mm laminated will give you a stiffy, and you don't need to be mesmerised by MDF (the god's gift to woodworkers). A few bracers like this hint will provide stiffies.. 1 (or 2) lamination of 3mm MDF (underneath) will put salt in the mustard (FIIK, what that means).
Laminates underneath may not look pretty, (especially from top down... doh), but where there is room underneath will only cost some extra glue.... and maybe dowel reinforcement if you are a  pussie  scardy cat.

If your first cut is too short... Take the second cut from the longer end... LBD