The readiness of a snake to bite while handled/manipulated really isn't of any value in determining whether or not it can do well in a multispecies enclosure.. I worked an exhibit that initially has beaded lizards, western diamond backs, black tailed rattlesnakes and a coachwhip in it for more than a decade. The diamondbacks wouldn't hesitate to attempt to bite me (or other keepers) when every we had to move them (using hooks) but could be tromped over by the beaded lizards without showing any concern at all.
I would however not suggest attempting to house a predator with it's prey species. If it smells like food to many snakes it is automatically food, there are reports in literature (including newspapers) about odd things snakes have been known to eat because it smells like food. These have ranged from reports of attempts to eat ceramic bowls (Kauffield, The Keeper and the Kept) to other items (like this x-ray that ended up in Ripley's museum Ripley's Believe It Or Not - BOADDICTION). This means the garter snake even if it can't reach the frog may be willing to ingest other materials that get scented with the frog.
The snake may indeed attempt to eat the frog even if the snake is too small.
As a final consideration, if the snake is too small to readily predate on the frog, have you considered that the frog may predate on the snake? Even if it can't swallow the whole snake, it may take the tact of other anuran predators and swallow as much as it can initially and then ingest the rest as it digests.
If you need a snake for the exhibit, you would be better off with a snake from the Storeira genus... there are some other possible snakes as well depending on the species that can be used.
when i was in elementary, middle, and high school i would keep a 29 gallon tank outside during the summers and stock it with tadpoles, mudpuppies, etc. I caught a baby garter, about 6 inches long and kept it all summer in another tank before letting it go. I woke up one night and found a big fat tree frog on my window and thought "o0o0 how cool! I'll make a tank for it tomorrow and just keep it with my baby snake for the night!" when i woke up in the morning my frog was gone and there was a huge lump in the middle of my tiny snake....
Let's not overanalyze this--the answer is no. Garter, Ribbon and Water snakes all eat frogs, as do Ringnecks and, of course, Hognose snakes. King snakes eat anything, and baby rat snakes love frogs.
Want a snake for a frog enclosure? Hmmn--I can't think of a better, easier one than a rough green snake. (Although I have seen zoo exhibits with arboreal boids and large tree frogs, and crotalids with large toads.)
Rough green snakes are insectivorous, and can be kept successfully with anoles and/or tree frogs in big, naturalistic setups. A bit delicate though, so I wouldn't handle one as readily as say, a garter or corn snake.
Storeria as I noted above are also a good option. I've worked with greensnakes in enclosures and they are a lot more delicate than many people think.... Unless you collect your own, they are often in poor condition and a lot of emergency work is required to pull them through. They tend to do best with daily feedings of crickets and wax worms (or other small caterpillars) while Storeria can be fed once a week on a decent sized earthworm (No red worms (Esenia spp) please).