I understand the potential, and likely, imbalances this will cause. It's a rough idea right now, but i'm a "jump in and try it" type, so I'll be play-testing this sooner rather than later
Heat
I've mulled over the idea of dividing a mech's heatsinks by 12 and applying that much heat dissipation per phase, but I think that is going to complicate things unnecessarily. I'll start with heat calculations at end of turn -- if you fire your PPC twice, you deal with +20 heat. If you fire your AC-5 five times, that's +5 heat, etc.
I -might- look at putting a heat limit in place to prevent mechs like the Awesome from doing a 3xPPCxTwice in one round Deathblossom attack. Not sure how much -- maybe twice your total heatsinks. I'll decide that after some testing.
Range
High heat weapons like the PPC and ERLL are definitely going to suffer, but I'm also toying with the idea of increasing the range of all weapons by x1.5 or x2.0. This will mean the long range weapons (which are generally high heat) have more time to deal their damage before the short range brawlers can get in close and apply a barrage of rapidly fired AC20 rounds to the mix.
Gauss Rifle
This weapon is probably my biggest concern using Heat as a firing rate limiter. 6 shots per turn? YIKES.

Perhaps we need a firing rate limiter besides heat...
Gunnery Skill
Along with Heat, Gunnery skill could be used as a limiter on how often you can fire weapons. For example:
- Green pilot with 5 gunnery can fire every 6th phase (he requires 5 phases of "reset" between each salvo). So he might fire on phases 1 and phase 7, or on phases 4 and 10, etc.
- Veteran pilot with 3 gunnery can fire every 4rd phase (he requires 3 phases of "reset" between each salvo). So he could fire on phases 1, 5, and 9, or on phases 4, 8, and 12, etc
- Elite pilot with 2 gunnery could fire on phases 2, 5, 8, 11
So while a small laser could theoretically fire every other phase (1 Heat = 1 Reset phase per firing), the pilot's skill would be more restrictive and reduce small laser spamming.
On the flip side, even though a pilot might be able to fire 4 times per turn, his PPC would only reset once for a pair of firings in that time frame.
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Wow, lots to test, now I just need some time to do so. I'm excited to see how some of the "flavor" mech designs behave under these rules; you know the ones with one LRM5, one LL, an SRM2 or a small laser -- random, headscratcher weapon loadouts that are really lackluster in CBT could be much more interesting.