That "flexing" is a real issue, I suppose. I regularly create inserts
out of "whatshandy," and currently am using some plastic salvaged from
a store display or a sign shop dumpster (can't recall which at the
moment).

I guess I'm not the "craftsman" you need to hear from as I regularly
settle for tolerances glue can fill and creative joints can disguise
and have yet to make a fancy-ass piece of furniture. But I have
successfully employed a variety of materials to create TS inserts and
would, in your case, heed the advice of the fellow suggesting
laminating a stiffener to the underside. Use the piece you have as a
template, and cut another 1/4" or so - (measure the widest part of the
"lip" on your TS to determine how wide) smaller all around. Line it
up, glue it up, insert and raise the blade. That should take care of
the flexing issue for you.

Mean while, when you go shopping, cruise around behind the stores and
peer into those large green dumpsters. A wealth of materials can be
found. My chicken coop nesting boxes have a floor and read wall of
easy to clean while plastic Bi-Lo signs (red on the side one can't
see) with bright red 1/4" dividers made from some plastic divider
materials from a Staples display. Those chickens will be so happy!