I just need a box bottom. Do you DIYers think a bottom made of foam board (like home made inserts) would be strong/sturdy enough for a box bottom?
Absolutely. Just today I took a PnP game housed entirely in a foamcore box to my local gaming group, packed in a heavyweight shopping bag with, like, six other boxed games of various sizes. It survived the two car trips, twenty minutes traipsing across town and being slung under the table and behind the bench chairs at the pub with no problem whatsoever.
I'm in the middle of editing a video on making full-size game boxes for PnP games, and the middle section is exactly this - making a box from nothing but foamcore. In my opinion it's a great material for home box making.