I'd be concerned with long-term stability (warpage) of appleply. I've seen plenty of 10+ year old cabinets with warped appleply/europly doors.
All our high end shops use mdf core for doors.
You could use a thin oil finish (like Watco) or have them lacquered after milling. The problem with factory finishing is avoiding scratches while you do the milling.
If you want to save yourself some labor, there is also a company that makes doors/panels for Ikea boxes with sequence matched veneers in any species you want. Semihandmade