FJ,
I did a nearly identical transom rebuild one yr ago, cutting the outer skin, similar to what you've started. I used 3/4" Coosa for the core, but the style of rebuild was pretty much what you've begun.
You may want to consider removing a small portion of the top cap on each side, so you have sufficient clearance to "insert" the new core from the top, as a one-piece unit, rather than in pieces...
To remove the old plywood, I purchased a cheapie ($39) 14" electric chainsaw from WalMart, then used it end-wise to get between the remaining portion of the inner & outer skins.
It had plenty of power for removal of the old wood, but not enough to damage the hull glass. Did a fine job of both cleaning the old wood, and roughing up the remaining skin, to allow for good adhesion, after wiping everything down with paint thinner and acetone to remove any traces of chain oil.
I laid up the new inner & outer skins, on a home-made work table ($45 in materials for the lay-up table), then trimmed and installed them after the core was inserted and set up.
Just some thoughts for you to consider...
Good Luck
PS:
AMEN to visiting with Warthog. His X-Shark is one VERY fine piece, and he & Lydia are excellent hosts. Hope you enjoyed meeting 7 visiting with them as much as I did