Looks like that's all of them. The typical number of even digits of n grows like a constant times n, so you need some very large deviations from t

I'd conjecture the number of powers of 2 with exactly m even digits is finite for all m.

This is equivalent to asking: how many powers of 2 are there such that all digits except the leading digit are 5 or greater?

The powers of 2 with a single even digit are just those double those numbers (i.e., the next higher power of 2).