Microclover pansys strawberry thyme lavender oregano tarragon or even using annuals in succession like beet carrot radish daikon things that can break earth and put aggregates in the soil so its not so hard clay like or anaerobic. Dont use mint way to aggressive its great around trees but i dont reccomend it around low shrubs they will choke them out. Also consider self seeding prolific usefull plants like borage or even using a prolific annual like ground cherry it really depends even nasturtium and when it come fall time just leave the annuals in and they will be a blanket and fodder for worms to continue to make pathways in the soil.
For groundcover around blueberries, strawberries (both wild and domestic) are a great choice. You could also do creeping raspberry (has little orange rasberries) or lingonberry. In shadier areas, you could do bunchberry. Pansies are also a great choice, like Jordan said. The slugs are all of mine, though. You could also do sheep sorrel--it doesn't seem to mind it wet. It does spread, but it has a lovely lemony flavor. I also like dandelions under my blueberries, as the flowers bring pollinators. You could also probably grow chives &/or green onions with the strawberries. Those all seem to do well with my wet, pacific northwest soil (though mines gravelly loam, not clay--you probably live south of Olympia, right?) All those plants could also grow under your trees.
Jordan has a lot of fantastic suggestions, for the non-blueberry areas, too, and I ditto all of them. The radishes will get tall unless you chop/harvest them before they bolt. BUT, at least in my soil, they self seed, and so I never have to plant radishes--they plant themselves! Nasturtiums are beautiful, and also often self-seed, and can handle a lot of weird soil and weather. Another edible ground cover that self-seeds is Salad Burnet. The young leaves taste a bit like cucumber. Oh! And parsley is great, too. I planted mine once, and it's self-seeded for years and I've never had to plant it again. You could also plant daylillies--not really a ground cover, but they'll do well there and are pretty.
I also don't think you'll need the groundcloth under the blueberries. Mine do fine with just a watering every few weeks, unless it's really hot. That was when they were young, too. I probably don't need to water at all, now. Also, adding a thick layer of tree timmings/mulch/bark/woodchips under your blueberries will be great for the blueberries, hold in all that winter/spring rain, and keep down weeds. It also looks nice. AND, the strawberries will grow merrily in there, too.