I am not sure at all that I understand what the two large holes are for (I don't think I have seen your rocket design).
Are these for smaller motor-mount tubes to pass through this bulkhead?
Is there some other reason these holes need to be this large?
Can you use three or four smaller holes, and leave more material in the portions of the bulkhead that 'bridge' between the holes?
If you must have these two large holes, you can improve the overall bulkhead significantly by using the better grade of plywood that wwattles suggested. Also, check out the 9-ply you said your uncle offered. If they are nine plies of about the same thickness, that plywood would also be better.
For a bulkhead with two large holes, instead of installing your U-bolt across the middle, try adding a stiffening web that extends from top to bottom on both sides. Attach a smaller U-bolt up near the top or down near the bottom. Or for an even smaller footprint, get someone to tack-weld an eye-bolt (so the perimeter of its 'eye' is completely closed) and bolt it into a corner.
Your broken bulkhead looks like it was made from 'veneer'-type plywood. These have very thin surface plies and a few, thick middle plies. They are NOT made for structural purposes, but for appearance (the surface ply is usually supposed to be a pretty wood). For model rocketry, look for lots of plies, low rates of gaps or cracks within each ply (peek at the edges of the plywood stock to look for ply quality), and good quality wood used throughout (not some thick layers of cheapo wood to fill up the middle). Home improvement stores in my area sell a style of plywood that is very strong and excellent quality but is quite pricey----I think it is called Scandinavian or something like that. One 4x8 sheet of that stuff would last you a lifetime of bulkheads.