If you're not married to flying into/out of TBS, you can always do one of the discount lines out from Kutaisi. They should have some flights to Europe at decent hours. From Kutaisi, you can take Georgia Railways to Tbilisi or Batumi--and from there, wherever you might want to go.

I myself have never flown out of Batumi, but my MIL had to make an emergency departure out of there in the late afternoon, again with TK.