A .055 fender flatwound G string is massive. Don't be afraid of tightening the truss rod enough to help the neck cope with these strings, because if you don't tighten it enough (while keeping the strings loose), the relief gets too large when you tune up, and the action gets way too high, and I'm afraid that keeping the neck in such a position for a long time might cause permanent neck warpage. It should stay rather straight.
The "rule of thumb" saying you should never tighten or loosen the truss rod more than 1/4 revolution per day is not applicable when putting on strings with a very different tension, and those strings are very high tension. Tighten the truss rod enough directly so the relief gets right immediately. Read more about relief in the setup subforum here on TB.
I've had Steve Harris super-massive high-tension strings on my jazz and replaced them with Pyramid Golds, that are much lower tension strings. No issues with the neck, you just need to adjust the truss rod accordingly and you're fine...
