Since you're military you can join NFCU, if you haven't already, I'd recommend joining and trying thr refi there. A relationship there may help with your home financing as well, they do both VA and conventional.
The difference from 5 to 6 inquiries isn't that big a hit. I wouldn't get so wrapped up in driving scores that you do things that don't make sense. Right now you're probably one "tier" below best mortgage rates, the best rates come into play around 720.
