It figures that the only problem is in the most visible panel.

I bet your “veneer strip in a shallow rabbet would be virtually invisible.

If it eats at you too much and you decide to re-veneer the entire panel, the iron-on method might work. It’s the only method I’ve ever used, but it was always with two-ply (i.e., thick) veneer. I don’t know if the heat would cause splitting or checking with raw or paper-backed veneer. I’d definitely try a test run.