Has anyone ever tried rebottling vermouth in smaller bottles to keep it fresh for longer?
No, but I usually buy it in tenths (375 ml) for that reason. I should explain that I just occasionally use Vermouth for drinks, opening a fresh bottle if possible. But for decades I've normally kept screw-cap-closed bottles of it on hand in the refrigerator (along with various other fortified wines) because it is useful in cooking. (Just used some to soak the saffron for a Risotto Milanese
in fact.) I find that it loses a little sublety, but doesn't show gross oxidation even with months of storage.
I should explain I deal with table wines regularly in large variety, frequent blind tastings, etc. After many years of experimenting with storing leftover wines, trying various gimmicks and ultimately rejecting them as more trouble than they're worth, the usual practice for partial wine bottles is to seal them up tightly and refrigerate.
That last factor is more important than anything else (slkinsey and others acquainted with thermodynamics will recall all the "kT" factors governing reaction-rate exponents) because a little reduction in temperature generally means a big reduction in chemical
reaction rates of all kinds, including spoilage mechanisms. (A popular topic online among wine geeks who are also professional chemists.) For tables wines, usually the leftovers are used within days -- allowed to come up in temp. a bit before serving, of course, for aroma and flavor -- and how much this alters their flavor depends very much on the wine. After hundreds of trials I've found flavor and subtleties often remarkably intact, and rarely is there any serious flaw after a few days.
Fortified wines are a different story, lasting much longer, in fact some (like Sherry and Madeira) are, so to say, pre-spoiled as a deliberate part of their production and flavor. ("Madeirized" is standard wine-taster speak for oxidation in table wines.) A well-sealed, refrigerated
375 ml Vermouth can be fine for cooking after months. It might disappoint a Vermouth connoisseur in a drink, but I'll confess to making well-received Martinis (real Martinis, with good gin) from it occasionally.