Perhaps it's just your wording, but you may have made the classic mistake of thinking work = FPS. I know for a fact that Section 8 (to take one of the sites you play at) has an FPS limit of 360 on fully automatic guns; using heavier BBs may reduce the velocity, but not the overall energy of the BB.
Changing your FPS really isn't always entirely necessary when you're using heavier BBs. Indeed, if you go in with the expectation of getting ~350fps with a .36 or something, then that's considered cheating the chrono and may have you thrown out.
The maths:
~360fps (which is what you should be at right now) with a .2 is 1.2J
~450fps with a .2 is 1.88J, which far exceeds almost every site limit in the UK for automatic weapons
If you were to use a .28 or .36 with an upgrades spring then you'd be putting out 380fps and 335fps respectively buy again, the work stays the same (at 1.88J). Sites tend to put the FPS limit in as it's easier for people to understand, but what they're really after is the energy of each shot being at the site limit. For Section 8 that's:
My advice: Buy those heavier BBs and crank your hop up to see if it'll cope. If it does then you're golden, and will only need to conduct minor upgrades. If your M14 is hitting 360-370 already, then that's the high end of being site legal anyway for fully automatics, and you should probably not take it any further.