Might not look the greatest, might not be the most professional way, but I would see about getting some of that fiberglass insulation like they sell on a roll, you don't need a whole roll so I'm not sure what to tell you, but 4-6 feet and just wedge it up in there.
The problem here is if anyone starts a fire and forgets the insulation is up in there, it might be better (if a bit unsightly) to simply stuff the cavity itself with this stuff and then close the glass doors, at least then any blind fool can see it.
That or do both, might be worth the 15- 20 dollars it would cost to buy a whole roll of like R-11 or R-19, get the unfaced stuff so you can just stuff it in there liberally. Yes you can cut it with scissors, probably shouldn't but it does fine.
Wash your hands afterwards
Might feel all fluffy and comfy, but it is fiberglass.