Erythronium (dog tooth violet)

Arum Italicum (Lords and Ladies) it may not flower well in shade, but I like the foliage better than the berries.

Anemone blanda and nemorosa are both good for shade, daffodils too. I think both of these are not eaten by rodents (no cage needed).

Is your garden as shady as you think? Most of the other bulbs might be worth a try if the area gets winter sun and the bulbs can grow before the leaves come out.

Scillas also do well in part shade and I beleive they are rodent proof too.

I assume you are talking about spring blooming bulbs, since you mentioned Snowdrops.

Taken from the link below.........

Will daffodils grow in the shade?

They will grow in the shade of deciduous trees because they have finished flowering and the foliage has begun to mature by the time deciduous trees leaf out. However, it is better to grow them outside the drip line of deciduous trees rather than under them. Also, deciduous trees with tap roots are preferable to shallow-rooted trees. Daffodils will not long survive under evergreen trees and shrubs.

Many of mine are growing, blooming, and thriving in the woods, even though they are planted within the drip line of deciduous trees.

Will squirrels and other rodents eat daffodil bulbs?

No. The bulbs and leaves contain poisonous crystals which only certain insects can eat with impunity. They may, however, dig up the bulbs.

Sue

Here is a link that might be useful: American Daffodil Society FAQ

Some of my favorite bulbs are shade bulbs:
Spanish bluebells (hyacinthoides) of which Excelsior is a superior form.
Arum Italicum Pictum "Marmoratum", a white variegated variety of this beauty.
Cyclamen hederifolium
Bletilla
Iphieon, Rolf Fiedler is a gorgeous blue (deciduous shade)
Check these bulbs for hardiness in your zone. I have no idea.

If your garden is shady because of trees/shrubs you might need to make planting pockets and 'import' better soil. Evergreens can be particularly difficult to underplant - holly, and conifers, rhododendrons and magnolias. It seems like no time at all before the soil is tired and the bulbs are shrinking. Even topping with compost can sometimes be not enough. The soil becomes full of fine roots, compacts, loses humus, dries out massively and is very slow to rewet because of the root thatch on the surface.

Hardy Cyclamen, woodland Oxalis (if it's not pesky in your area), [Clivia - must be frost-free] Iris gracilipes, some of the lilies, some of the fritillaries, trilliums.

If you have a precious sunny patch you might be able to use it for starting containers of bulbs which you could move to chosen sites for the flower effect - then move back again to finish the grow-cycle in the sun.