..We are trying to decide it we should bolt the hitch through the mat or cut the mat around the hitch....
IMHO Cut it around hitch. You do not want any flex on what the bolts hold. Too much additional shear forces could be added on the bolts by the mat allowing flex.
Now as a caveat to that... if your talking about a thin mat like 1/8th" then would think should be fine. The more above that you get..... the more you need to consider to cut it out.
Then there is water to consider along with grit and grim that can get under it to rub off bed paint
under where the mat is. If the mat helps it hold water then your inviting rust. Especially with those newly rubbed off paint
spots and... low and behold; holes drilled to mount it exposing nice fresh bare metal.