I have glued up panels for the bulkheads of my Guillemot using leftover strips. The strips started out 3/16" thick but I had trouble keeping everything flat when I glued the panels. After running the panels through my drum sander to flatten the panel the final thickness came out closer to 1/8". I know some people use foam but I would like to use wood.
Will the thickness of the panel matter?
Yes, the thicker the core the stiffer the panel will be.
I'm assuming you are going to glass both sides of the bulkhead panel before installing?
After it's glassed the panel will get a lot stiffer. If you are still concerned, just laminate another layer of glass on each side -it won't add much weight.
I use glass-epoxy bulkhead panels without a core in my boats and they are strong enough.