What would you say is a fast curing finish, that imparts no odor when dry? I'm thinking a water based acrylic poly, or shellac. This is a wood item for a friend's relative, who is extremely sensitive about almost everything. Since the wood is basswood, some ambering of it would look best, which seems would be a problem with acrylic poly. Maybe shellac first, then coat with acrylic? Or something else?
GF Enduro-Var. ticks off all your requirements.
You're on the right track, an acrylic finish or shellac will do it. The acrylic will almost certainly be more durable, and maybe easier to apply. The Enduro Var product that Carolyn mentioned is a very good one and should be given consideration.
EnduroVar is a great product; I use it extensively. However, it does have an odor that lingers for quite a long time, especially in closed drawers. It takes at least a week for the odor to completely disappear, which is about the length of time it takes to cure so maybe there's a correlation.
John