If I faced the same situation, I'd maybe do this:
Not sure why full-face contact is needed for the entire 4" height of the edge?
Top upside down, straight edge running the length, router with top-bearing long bit . Route the bottom [currently on top] of the offending edge. Route it back far enough so that it is narrower than the narrowest part of the top [which is now on the bottom].
Flip it over. Now route a new face at the top of the offending edge. It needs to be straight, normal to the top, and WIDER than the face left from the upside-down pass. I'd hope for 2" face - maybe 1-1/2".
Glue and clamp the expansion board to this face. Counterbore, countersink and drive serious fasteners. I'd maybe go for 1/4" x 6" SS lag screws. Now the expansion piece is in place.
This solution ain't goin' nowheres. Clamp to your hearts desire.
You could always get the auto shop creeper out and scurry along custom-fitting and gluing shims in the gap at the bottom. If you don't, within 90 days you'll have forgotten there's a gap, and within 9 months you'll have forgotten the entire adventure.
UNTIL you add a dog hole and find onea them 6" lag screws. So I might make a shallow saw kerf and filled with red paint to mark their location. Or a wood-burning pen. Something. Conversation piece.
PS - while describing this, I kept thinking of the classic MAD magazine take on The Poseidon Adventure. They called it The Poop-side Down Adventure. The entire bit had the characters arguing if they were going up to the bottom, or down to the top. You gotta be a certain age, I guess.
When I started woodworking, I didn't know squat. I have progressed in 30 years - now I do know squat.