I have put a pet door in an old solid wood door. We got lucky, and were
able to just remove a panel that was the perfect size.

If the "wood framed/steel faced" door is more or less solid wood, then
I guess I don't see why you can't put a pet door in it. Anything you do
(cutting into a door, wood or metal or whatever) is likely to be
permanent anyway. You can use an angle grinder to cut a rectangle in
the metal, then a wood saw to cut out the wood insides. Or maybe just a
sawzall or sturdy jigsaw to cut through both at the same time. A little
caulk when installing the pet door will keep out the rain and snow. of
course, the pet door will never really be well insulated.

You may prefer to just go through the wall next to the door. It would
be easier in some ways (no metal, no $500 door to ruin), but harder in
others (multiple layers of drywall, studs, sheathing, siding, etc.)

-Kevin