You want to rabbet the additional trim piece to allow it to sit flush on the existing Oak trim and also flush with the Melamine door panel...right?
I have 2 concerns ...
First, is the existing trim the same dimension off the Melamine on every panel? This would be great, so that there are no gaps when your new piece is applied.
Secondly, any finish on the existing trim will prevent glue from adhereing to the Oak, and Melamine is notoriously difficult to adhere to with glue. A contact adhesive or Liquid Nails works best for me in these cases.
There are 2 ways you can "frame" the existing panels, one piece applied at a time, OR make a entire frame assembly and apply it as a unit over the existing door panel. You may have to have different hinges if your added thickness interferes with other doors or the adjacent walls.
It's an ambitious project, but you may find a different solution...new doors?
If you have a table saw, I would consider running each panel edge vertically against the fence and trim off the existing Oak and up into the Melamine enough to make a new surface to adhere the new frame. ... also a bit ambitious.