[DesertRatTom]

Screw the thin piece to the top from underneath so you can replace it should it bend. The only time you are applying pressure with the clamps is when you have clamped a piece in place. Quarter inch BB ply should work fine, and is expendable

If your stile is narrow, you will want to be able to clamp another piece in front of it, with your clamp on, or spanning both the first and second piece.

I don't much care for coping sleds and learned to just use a square piece of MDF cut nice and square. I hold the workpiece in place with my hands, which is adequate since the end is also running against the fence,so it is automatically positioned correctly. If not, make a second pass.. The push block will be chewed up for an inch or so past the corner because I run it against the fence. Don't want to change the fence position or bit height between rails and stiles. I do use matched bit sets, so changing height defeats the point of matched sets.

Check out some of Marc Sommrfeld's technique. He doesn't use a coping sled and it's how I learned to use simpler methods.

If you are using matched sets, drop a grommet (not an O ring) into the collet so the bits bottom out at the same depth. I use only matched sets..

 

[DesertRatTom]

But nobody use 1/4" cross-cut sleds for their table saw. It's almost always 3/4" (now 18mm).

But a coping sled has a completely different purpose and requirements. If 1/4 raises the bit by that much, the 3/4 ply will require an extension, which the OP wasn't eager to pay for. It is a replaceable part of a shop made jig. BB is unlikely to bend, while cheapo Chinese ply might. Again, the only time clamping pressure is applied is when it is in use, and then it's supported somewhat by the table itself.