Well, 2 years later and I'm getting closer to actually implementing this idea. This weekend I mocked up the fans using double sided tape, wired them together, and tested operation using my switching power supply.
I had to remove the igniter from the mount because it was hitting the fan, but this setup seems to be best. I'll run screws into the grate to secure the fans.
I'll use the temperature switch from the squirrel cage blower to activate the fans, and a 12v AC adaptor I've got laying around. At 12v, 0.31 amps are drawn for just under 4 watts. These are way quieter than that noisy blower.
I will block off the sides using coroplast (maybe, unless I find something more heat resistant). It really isn't feasible to put fans on the ends.
Oh, that light in the ziplock bag is a WiFi switch so that the Chinese can control my fireplace using IoT. I've got Google starting the fireplace automatically during the morning routine. I had to become an expert on how the gas switches on and off electronically to get the right switch. Fireplaces that use a pilot light generate low voltage from that heat that is then used to open the gas valve when the switch is closed. I made the mistake of initially thinking regular house voltage was switched to activate the valve.