I wouldn't drive on a run flat due only to the fact that if you do, you will destroy the tire at road speed. I would drive on it only long enough to get me to the next exit on the highwAy, to a gas station, ect. Most places dint know how to deal with inserting/removing the run flat. ( it is a PITA), nor will they have replacement O rings for the split rims. (you will need the red ones for the HMMWV)
Other than that they are great tires, but heavy as heck. If having the HMMWV tires isnt important, you can sell them, buy CUCV rims/tires, and make a tidy profit. Plus, you have tires that are easily fixable/replaceable and cheap.
I would not use the fix a flat type stuff in ANY tire either, unless it was a last resort. It throws the tire forever off balance, and is incredibly hard to clean.
For purposes of getting the trailer 300 miles to home, bring an air pig or comprerssor make sure they are inflated properly (35 PSI is what I run mine at) and go. Make sure all lug nuts are tight. 99% of the time the nuts holding the rims togther are OK (not loose) but it cant hurt to check.
Also, do yourself a favor and take the 10 minutes to remove the cover. No need to subject it to road speed winds for no reason, and your MPG's will be higher.
Whats your plan for lights on the trip home?
As a general rule, 2 wheel trailers are pretty easy to recover. Not much to go wrong. Low tire pressure and bad bearings are the 2 biggies for potential problems when picking up from GL. Bearings are easy to check, drag it a few miles down the road and feel the hubs. really warm or hot hubs = bearing problems. Warm or cool hubs = good to go.
Also, I'd take the cover off for the ride home, maybe even the sideboards too. No need to drop your MPG's for nothing.