Somehow I missed the backboard part.
It ain't going to be worth it. You're just going to waste time and money. Use hardware, replace the backboard with something else, or if the HDPE/acrylic is thick enough for whatever it has to do, just ditch one. Just make sure it's safe.
Buying what you want is incredibly difficult and expensive. It's a bigger project than you might think.
I love HDPE, and I hate HDPE. I think that's going to be the case for anyone who doesn't work with it in a factory stocked with what's needed.
If you want to sandwich them with hardware, just like glass it makes sense to wrap the edges in channel of some kind (aluminum U, for example), use bolts and nuts, and drill the holes in the materials themselves a little larger than they need to be to allow for expansion if you're trying to "laminate" them together.
HDPE likes to move, and it will crack itself in the winter given half a chance if it's restrained. In the summer, it's possible for it to crack the acrylic.
It's not as bad as attaching them to metal, but they do have different expansion characteristics, and that can produce an
obscene amount of force.
You'll want to assemble them in a somewhat average temperature. IF you do it outside in the cold, you're starting off at an extreme. Do it inside at a slightly higher than average temperature you expect to account for solar loading in the summer.
The only reason more people don't use more HDPE more often is the fact that it's a pain in the ***, and you've pretty much got to heat it into a shape or screw/bolt it together, it eats blades, it eats bits, you have to cut it slow or it will melt...
Like I said, I don't know how it's put together, but I suppose it must be fairly obvious what's safe and what's not when looking at it. If you use screws, make sure they aren't sticking out the back in case something breaks or falls.
Or just screw the ****** together and hope it works.

You'll probably get away with it. The above is just all the things that factor into laminating two polymers when one of them is hard-headed.