XDSTEEL, there technically, isn't such thing as a Florida NON-resident license. You're living out of state simply makes it NON-resident. Otherwise, the license you'd get in the mail is no different than mine. They don't even put your address on them anymore.
Florida is pretty easy. Just go to this website.
http://licgweb.doacs.state.fl.us/weapons/index.html
I think you can download the application as well as the fingerprint card off the website and print them up. Or, I'm sure you can request the "packet" sent to you. Just be careful and read the directions carefully. There's a pdf format download of the instructions online. There's a couple questions that, IMHO, aren't very clear and can stump you if you don't answer them right. For example, questions 4a and 4b inquire about exemption from Florida's public records laws. Usually, that regards public service folks. Not difficult, just be thorough.
Simply fill it out, get it notarized, have your local PD or SO ink your fingers, write a check, stuff it all in an envelope, and mail it in. (suggest return receipt mail, that way no question they got it) I think they're running somewhere between 2 and 4 weeks to your mailbox these days. The 90+ wait times seem to have subsided.
As far as your proof of your training. I'm certain military training suffices, I'm just not up to speed on what documentation you need. Browse around the above website and give them a call. For a government dept., they're honestly not too bad to deal with on the phone...usually.
Good luck, and stay safe.
OH YEA....make copies of EVERYTHING. Send the originals...but, keep copies for your own peace of mind.