OK, I did actually take the phone and rule to take some pictures...actually, the shaft is 3/16", not 1/8" -- sorta' obvious after being awakened by reality--a 1/8" OD with the slot woulda' left nothing at all to speak of.
I've got pictures of another shaft that is still intact as well as the stub end of the remainder in question as well as the knob -- you'll recognize it as quite common but I just couldn't find an online picture for one. It's just under inch long and 1/2" OD at base with the formed flats for gripping...
That wipes out my Nicopress idea; know I have nothing that big but makes finding something to use for starting point easier, I'm sure...
The existing opening in the oven plastic cover is right at 3/8" so a wall thickness of <=1/16" would still go thru and be stout enough to take a good squeeze. I suppose it wouldn't be totally impossible to remove the clock and manage to even drill it for a tiny roll pin or the like if wanted to get esoteric...altho I don't have much of a set up for precision work--farm stuff doesn't tend to need much precision; everything I have is basically for big, rough work. :)