Something like this is kind of what I was thinking. I've tried something similar before for a related experiment and had mixed results, I guess because displacing the air isn't that efficient.
I thought about the problem some more and think another option is to use our vacuum oven. Since vacuum is, in a way, an inert "atmosphere", and I have no solvent to worry about, so it may be easier than trying to displace the air with an inert gas.
Thanks for the ideas!