Well, the way the shields anchor, they hold the tube in place. I think you could go without them from an electronic perspective, but playing vibration and travel shocks could allow the tube to work free.
If you want to try those tubes, you'll just have to keep popping the cover to make sure they're well seated all the time.
Another thought might be to get your hands on another set of the tube sleeves. I suspect that if you used the right tool (dremel cutting wheel, maybe?) and sliced the tube 80% or so of the way up the sides, you might be able to get them to support a tube that is slightly too large. Don't know for sure as I've never tried it, nor do I know if it would be a problem for the tube to be held that snugly.
Finally, I haven't tried it in the F-1X yet, but the JJ Tesla long plate tube (ECC803s) sounds very round in my Fender pre. I think I have the short plate in my SWR Baby Blue II right now and it also has great tone. These tubes are in the $10-12 range at many dealers. In each case, they replaced a GT tube and I preferred the result.
Good luck!