I'd suggest reducing Rapidfire's GCD reduction to 2 seconds instead of 1.5 and double the number of weaponskills it affects. That would help with the ping/weaving issue while retaining the effect it has.
I'd also suggest leaving Barrel Stabilizer in and adding a mechanic to Overheat to reset heat to 50 instead of dropping Gauss Barrel when done correctly instead of the penalty. One idea I had is simply to make the goal to reduce your heat below 50 before your wildfire finishes. Allow Hot/Clean/Split/Slug Shot to generate 10 heat at all times, then make Cooldown reduce it by 10 outside of Overheat and 30 during it. Ammo would still reduce heat generation too. Finally, allow heat to go above 100 during Overheat, then after overheating, add a debuff to discourage overheating again too quickly, similar to the Turret debuff from Overdrive.
By doing all of that, you then create a situation where Barrel Stabilizer is used to correct mistakes or deaths without majorly impacting the rotation, and a reward for correct heat management that's rather easy to do (3 Cooldowns or 2 and proper ammo usage)
I agree with what you wrote otherwise.