There are several good thread on the subject. The answer is maybe. First of all when you say bush cherry your taking about sour cherry flavor, not sweet cherries. There are not bush sweet cherries. Second does it have to be a good cherry or an OK cherry.
Meader cherries are quite OK cherries. They have been no spray for me.
Jan, Joel, Joy Dr. Meader cherries - General Fruit Growing - Growing Fruit
I have yet to taste Nankings but they are often grown from seed so they range from OK to good. The forums also agrees they are rather temperamental. One Vendor Edible Landscpes is noted for having 3 white nanking cultivars Edible Landscaping LLC the original since 1979 and they are out of afton VA. So that should be very simular to your climate
Then you have Sour Cherries - University of Saskatchewan Fruit Program | University of Saskatchewan (usask.ca) Dwarf Cherries. These have similar genetics to sour cherries eg, 3 sets chromosomes from prunus fructosos and one from prunus Avium (sweet cherry), sour cherries have 2 sets from p.fructos and 2 from p.avium.
There bred with colder Climents in mind but do well furthur south also
Unreleased University of Saskatchewan prarie cherries we want & what we know about them contrasted with romance series cherries - General Fruit Growing - Growing Fruit
Finaly, does it have to be a cherry? Plenty of fruits taste like sour cherries.
Goumi’s for example grow well just about everywhere.