Warning: Some of these solutions may actually be or become problems, due to the invasive nature of most of my suggestions. I'm just assuming you might prefer some of them to garlic mustard. Use my suggestions judiciously.
Clover might work reasonably well.
Creeping Charlie would probably work great. It is a weed and hard to get rid of, but it's perennial, evergreen, grows in sun or shade, is edible, and it doesn't have thorns. It also reseeds, a lot. You could probably walk on it if you wanted to. It looks nice in winter. It grows well in western Idaho.
We've got what seems to be a hybrid of creeping Charlie and catnip that would probably work, too, if you want bigger, more fragrant leaves with a similar growth habit. I don't know if the hybrid is a perennial or evergreen, though. Catnip isn't. But, this hybrid grows back after you pull it up (perhaps even more successfully than creeping Charlie does). Catnip doesn't grow back after pulling it up, typically, unless by seed.
Chives should work well. They grow well with little attention, sun or shade. Chives are perennials and reseed a little, too. Chives aren't aggressive, exactly, but they tend to maintain their territory and expand gradually.
Spearmint would likely work, if you planted enough of it. Peppermint won't reseed like spearmint. So, that's why I recommended spearmint. They're both perennials that spread through rhizomes, though. Mint will grow in the shade.
Calendula, chamomile and sunflowers reseed like crazy in western Idaho. You might consider those. I'm not sure how they do in the shade.
Arugula would potentially work, too (if you seeded it thick), although you'd have to count on it reseeding with the winters that you mentioned. Eruca sativa isn't a perennial, but Diplotaxis tenuifolia is, down to zone 6. However, at least Eruca sativa is easy to pull up, though. So, the animals might just eat it roots and all. The roots are tasty.
For the bonus you mentioned, you might try Chinese Lantern Gigantea, or any variety of Physalis alkekengi. It's highly invasive and once successfully grown out should give the garlic mustard a run for its money, if what the reviewers on Dave's Garden say is true (but you may never get rid of it). It's a perennial (down to hardiness zone 3 or 2) that spreads rapidly through deep rhizomes. It reseeds, too. It produces edible-when-ripe ground cherries full of vitamin C. I think they're poisonous when unripe, however. The husks are poisonous, I think I've read, too. The husks around the berries are ornamental. I don't know if the leaves are poisonous or if any animals will eat them.
Also, if you can get thornless blackberries established for a couple years, they should take care of themselves (unless you want to prune them) and spread quite well after that. I've witnessed that. There are thornless blackberries. We have some. You might think blackberries are more for wet areas, but they do well in dry areas (sun or shade) once they get established. They'll need watering regularly the first couple years, though, but you shouldn't have to water them at all, after two or maybe three years.
Black currants also grow very well in dry climates (and spread through growth and seed). They're wild here. I'm not sure how they do in the shade, but it's worth a try.