I think you might be safe to apply just the Heat generated by the initial Order. It's a free action and the target still has to be close enough for the mech to move to base contact.
It's not like Kisho's Anticipate, which gives Kisho an extra Order with all the Heat and Pushing that applies to it.
ANTICIPATE (Optional) Before giving this unit an order, you may give it a move order, which does not count against your order total for the turn. This unit may not use the run option during this move, and it may not be in base contact with any opposing figure at the end of the move. This unit gains 1 heat after resolving the move. ...
Kisho can generate up to four Heat with Anticipate, Push, Assault, and Pulse, so having several coolant trucks is almost a necessity for him. That's not counting any other Heat he might have on the dial.
Julian can add up a Push, Assault, and Pulse (or PPC Cap) and still get that free action without adding more heat. Though I will have say the odds of not shutting down after resolving the Order is not good, and the Heat is applied before getting that free action. So there are plenty of things that can make the Special Ability useless.
1. The Target is eliminated by the Ranged Attack and the Ability is not needed.
2. Julien's mech shuts down and can not use the Special Ability while shut down.
3. The Target is too far away or has a ring of infantry that can prevent base contact with Julian's mech.
It is a high risk Ability, so the pay off should be just as high. With a cascading heat dial, the Enforcer can't be too aggressive when it's not on the green click.
GC