If the tech upgrade Threat Tracker is triggered when the opponent's ship activates during combat and you use the ability to perform a free barrel roll and you have boost in your action bar, can you then use PTL to boost as well? (Assuming you also have the Elite PTL skill).

Yes. Assuming you haven't used it already this round.

Remember, you cannot perform the same action more than once during a single round, even if one or more of the actions are free actions. And you cannot perform actions while stressed. Other than that, you're good to go if you have boost and barrel roll on your action bar.