Is it feasible to solve this subset cover problem with SAT solver?
Optimization with SAT is usually referred as MaxSAT instead of Min SAT. In particular, I suggest looking for solvers for "weighted partial MaxSAT", for example MaxHS and RC2.

The problem size you have is fairly small in the context of modern MaxSAT-solvers, so yes it is worth trying. There are no guarantees that the solver will work efficiently, and it is very hard to predict if it will work efficiently, so the best thing to do is to try.