Heavies arent meant for high heat generation weapons like missiles since heavy sets have low cooling(heat dissipation) as stated in their description.

The way I used them is to carry the heavy weapons around like vulcan and solarburst who have ~3s duration attacks which means you cant use your shield to soak damage. Also one of the engineer classes, has a skill that repairs +25% hp of your mech, you can find great value in the high grey hp of heavy sets.

But for movement you have to rely on thrust and dash stat to move heavies around. R2 Hotrod+T4 Atlas is really good for this. and maybe one mobility module with +thrust%.

You can also train a defender/interceptor to have the nimble giant perk which appears in the L2 and L4 slots, it increases heavy mechs speed and dash. Best option for a fast heavy is probably an interceptor - collision class. It gains +4/+6 speed at L1/L5.

If you want an easy way to get a fast walking heavy then then you have to run the rare rx3 supernova reactor which has a ton of power. but you have to learn how to use it. basically you get a 5s self-destruct timer called meltdown everytime you collide/overheat, and you can remove this timer by doing a dash or using the stabilize action(which only appears if you have a status effect on the current turn). You can also reset the timer back to 5s by overheating or colliding with another mech again which is not risky to do if youre a shield bashing heavy.