For wqhatever it's worth, there are really good PSA veneers on the market today. If you cleaned the old panel with either mineral spirits, or paint thinner, make sure it's dry.
Then use a veneer hammer and apply the new PSA material over the old bulkhead.
I won't promise it'll work, but I've done several things this way in restoring antique furniture and have had very good luck with the process. If it does work, and I think there is a very good chance it will, you've saved yourself a lot of work.
If it doesn't, you can always replace the bulkhead with 1/2 or 5/8" ash plywood..