If I was going to carry mostly for snakes, 22RF birdshot would not be a choice at all, not even a last choice. Birdshot in several 357MAG's, a 44SPL snubbie for a time, but the 22RF's I carried for same shot bullets, and doubled for small game. Any of several 22RF revolvers long ago not as compact, S&W kit guns, Ruger Bearcats, etc, an Astra Cub 22Short for a time... the last one bought for the use a Beretta M21A Bobcat 22LR (they are also available in 25ACP).
That one would be my first choice for the OP's want of very small and very light 22RF (2.4" BBL, less than 5" OAL, 11-12 ozs) with steel barrels/slides and alloy frames. They are very durable and reliable, single or double action, have very easy controls easily useable with gloves on, have very good safeties, are plenty accurate for even small game at surprising ranges, have the advantages of a tip up barrel, 8 shots is plenty an extra mag if you want more and can be bought brand new below the price limit, for less used. Extended pinkie mags are available as are other grip choices, gloss blue, matte or stainless finishes. Add a cheap nylon belt holster or stick it in a pocket and you'll never know its there until you need it. It's only drawbacks is it is very loud, not a plinker without ear protection; and some folks have said they are picky about ammo because they are a true blowback. Never found that to be the case with mine at all, it shot everything well and reliably from cheap bulk pack Federal AE's to the super velocity ammo's. I never had a misfire, jam or failure to eject with it. Mine actually got used way more for cook pot fodder as most rattlers got a free pass. A couple of Mojave's that had me cornered and wouldn't back off, continued to advance aggressively as they usually do and I couldn't get past them without stepping over them, so they didn't fair so well. And a Western Diamondback that had my wife cornered and dead to rights ready to strike inches from her feet. One shot was all it took to behead it from maybe 8-10 feet away to the side. She fully trusted my abilities with it, but the muzzle blast still made her jump. Second time I had to rescue her in the same situation, the first in the early 70's with her 40'-50' ahead of me on a hiking trail. I was packing a Colt New Frontier Buntline as it was on a tree squirrel hunt, but it got it done just fine.