I'm suspecting that, because of your zone, your bulbs are planted reasonably deeply. If it is so, and if you don't mind lifting them for winter - would you consider growing tuberous begonias? They like some sun but also do well in dappled light.

You might want to try liming the soil below the tree. About 8oz to the square yard. It will sweeten the soil and, over time, help the clay to become more workable.

For foliage after flowers - Epimedium might be useful. You can get different colours - pink, mauve, cream, yellow, orange - and several have nice autumn foliage, too.

Bergenia is another possible. If you had the white-flowered one it would probably fit with the spring bulbs.

If the light below the apple is dappled, then Clematis might be possible, too. C. integrifolia, or heracleifolia, perhaps. They're both groundcover/scramblers and they have summer flowers.

And, however trite - there's Impatiens. (Go well with begonias...)

We have a dwarf apple tree is our yard (it has gotten pretty tall though) and had trouble getting anything established underneath it except, of course, the dreaded Bishop's Weed.

First of all, don't let Bisop's weed take over. Yes it will thrive just about anywhere, but it is an absolute nightmare to remove, especially once it has established itself. If you don't like it, get rid of it early and completely. If you do like it (for the foliage or whatever), at least contain it within a buried pot of something like that.

About the area under the apple tree. It took me a long while to get rid of the bishop's weed, and then the soil needed a lot of amendment because, like you, it is very clayish. We have selectively pruned the tree to let in some more light. Not too much off the height, but we spaced out the branches a little so that light passes through. Take out branches that don't produce any or much fruit and just prune them off. Take a little off the height of the tree as well because it is healthy for the tree. Also, take off some of the smaller branches which grow out of the main ones to create more space. In doing this, you will create breathing room for the tree branches and allow some more light to pass through to the ground, which will let you grow most "shade" plants, while not harming the tree (pruning is actually really healthy). You'll prune either once a year, or once every two years depending on how quickly the tree puts on new growth. Our apple tree will recover from a drastic pruning within the year, and surpass its previous height, so we prune every year.

We grow daylilies, hosta, astible and other standard shade plants underneath there now. I don't think apples falling hurt the soil. They do squish the plants if they fall right on them, but the plants tend to bounce back! They add organic material to the soil as they rot which i can only imagine is good. I've never seen anything harmful come of it. Except, of course, that the fallen apples can attract more bugs.

Ripening and rotting apples produce ethylene gas, which can affect the growth and flowering (and livelihood) of many plants by inhibiting the closure of the stomata, thus allowing excessive water loss from the plant. Some plants have a gene that makes them immune to the effects of ethylene. Recent work in this field, to implant that mutant gene into various types of plants, has produced some popular long-lasting, flowering types for the floriculture industry. Bishop's Weed (Aegopodium) apparently is one species of plant that has the protective gene.
Rapid removal of dropped fruit and good air circulation might be helpful in reducing the effects of the gas.
That sweet, ether like odor from the rotting fruit is inflicting damage upon surrounding plants.
BTW, all internal combustion engines produce ethylene gas, in addition to carbon monoxide. Vegetables and plants shipped long distance by truck, have to be protected, by a neutralizing agent, from the detrimental effects of ethylene, as do greenhouse grown plants, when heat is provided by gas furnaces.
Cigarette smoke also is a source of ethylene, another good reason to give up the habit!
Some trivia that you might find useful, or not!
Rb

I've seen apple trees with hellebores planted underneath and it was beautiful!