I did similar experiments with my PID oven, mine is a multi zone controller or whatever that can turn on a cooling system, but I have not done it. I thought to put a blower fan on top and channel it in, but I think its good enough when you open the door. you could just put a dryer flap on it so when you turn on the fan it opens the flap and cools the oven quickly, you need 2, one to protect the fan, and the other to open the oven.
but its a fair bit of sheet metal work because I don't want hot air shooting out of the side of the thing, I would need to make a diffusor system so when I leave something on top of it, the damage is not catastrophic
if you want a 'nice' mod, you need to do this on top of the oven, and figure out the correct springs, and install duct on double wall thin sheet metal.. its a fair bit of work.
I used a expensive SSR for mine ($60) because the parts I put in there can easily be a couple of hundred dollars, so its pointless to go with china reliability here, since I am not drying painted door handles.. personally to me a ruined pizza bagel is enough of a problem I don't want to go to china. Seriously when you pop something complicated into the toaster oven chances are you are tired as hell and going to be mega pissed off if it burns food lol..
They are useful also for doing things with varnish, I varnished my transformer in it from a HP supply to stop buzzing, after vacuum impregnation, since you can set the temp precisely to datasheet parameters.
I am super interested in high quality pizza bagels though, I wonder how much customers they lose because of bad thermal control. I have no idea how such a good food got a reputation along the lines of a pop tart.
Also getting a steady flow inside of the thing to cool it down nicely without thermal gradients is gonna be hard, because if you spot cool it with air flow you can break parts. I think basically opening the door is the safest option. They sell a nice D-cell powered fan for 25$ you can set next to the oven, open the door and turn the fan on to blow into it, it lasts a long time on a set of D's, does not have a cable and you can put it far away enough to cool the board gently.
What might be best is another chamber called a cooling chamber, so you take the board out when its solid and put it in there with something designed for cooling rather then heating!
I think I will use an AUX on the pid controller to sound a buzzer and turn on a light when it detects the solder is solid and then make a cooling box some how.
Keep in mind too those elements will last shorter if you cool them fast, so its probobly better not to even do cooling in the oven, and close the door after you take the board out to cool off.