I've found that training in a traditional martial art offers an excellent path toward doing exactly what you describe--activating thoughtful and emotionally appropriate responses while calming the instinctive fear response. There are two advantages I see to working on this type of thing within a traditional martial context.

One, old school martial arts rely heavily on kata, scripted patterns of attack and response. A real attack is going to happen quickly, without giving the attackee a chance to change his or her response pattern. However, when you're learning a kata, you begin slowly, giving yourself time to find and strengthen a thoughtful, calm response. As you get comfortable with the kata, the speed of the attack increases until it matches the speed of a realistic attack--but you retain the ability to respond in a calm and measured way.

Two, traditional martial arts have a lot of formality around them. Often, you bow or acknowledge your partner prior to attacking. This sets the practice world--the world that exists within the confines of your class--apart from the every day world. It's like overcoming arachnophobia by looking at pictures of spiders instead of real spiders. Then, as the intensity of your training increases, it's like watching a movie of spiders or looking a spiders with a VR headset. Eventually, when you come across a spider in real life, you're okay with it and you know what to do. The amygdala, having gotten habituated to a new response pattern in simulation, no longer overreacts in reality.

My (admittedly brief) experience with modern martial arts and MMA is that they throw you in the deep end more than I liked and they assumed that all I needed to learn was the mechanics of how to strike and block. The traditional martial arts I've studied have pushed me, but have spent a lot of time teaching me to retrain responses ("don't forget to breathe," "keep moving," "own the line," "slow is good and good can become fast," etc etc) in a way that has improved my overall experience in life.