You can probably start off as a Senior Rad Protection Tech. if you stick with dosimetry type stuff. Otherwise, you could work as a junior tech and become a senior in far less than the normal 3 years - maybe 6 mos.
It's a bit of a stretch to come out of the Navy and be a supervisor of anything right away. Things are just too different here. We have REAL radiation and contamination here. Dosimetry is a lot more complex (though not really more complicated if that makes any sense to you).
I know of a few corpsmen who work in Health Physics/Rad Protection, especially in dosimetry. Of course, in the old-old days the HM did radiation surveys on the boats too.
As far as employers go, there are lots. No need to limit yourself to nuke power plants. There is DOE, Radiopharm companies, hospitals, dosimiter manufacturers, and some other really cool places. I once interviewed at a place which manufactured a device that clears restenosis in heart patients who had stents in their arteries. I forget who they were, or what the treatment was called, but the device stored a row of small Sr90Y90 sources and pumped them up through a catheter to the stent.