Heat is there to remind you that there are bigger, structural consequences for your actions, as opposed to immediate or personal consequences. In a game based on Star Trek (as well as others), what would be the bigger consequence of blowing up an enemy ship instead of negotiating with them? Who would be upset?
If you don’t want a larger structural pressure on the party, I added the concept of Slights and Grudges, which describe interpersonal consequences. The setting of Asphalt and Trouble is very spread out, so I added the mechanic to bring the focus back into a smaller, tight-knit group, but it could work for your purpose as well.
In a nutshell, when someone in the crew does something that would upset an NPC, that NPC is Slighted. If the character or crew does not make amends with that NPC by the end of the next downtime, the NPC has a Grudge, which requires a very significant effort to remove. That NPC will now go out of their way to interfere with the crew, acting as a new roadblock and challenge.
If you want, rather than counting abstract Heat, you could have the GM review the NPCs affected by each score at the end of the score, and issue Slights based on the same kind of triggers that would normally drive Heat. Entaglements could include options to bring NPCs with Grudges in, or become harder the more Grudges you have.