How can I see around my own muzzle flash?
Solution 1:
The Exo is a suppressing fire support unit, when you are using the minigun and not the siege variant, that does better supporting other softer units than by itself. The minigun is not terribly accurate making aiming it at specific body parts (like the head) difficult. Also, Assaults and sniper units are the paper to the Exo's rock. These classes can deal a lot of damage very quickly and the weapons they use tend to be more accurate, making aiming for the Exo's "softer" head easier. For these reasons, the Exo is best used en-mass to create a wall of bullets or along side allies of different types. Both of these options make the muzzle flash less of an issue because the direction/location you are firing at becomes more important than any specific target.
Solution 2:
The FOV (Field of View) slider is likely the only thing that is going to give you some help with this issue in terms of options and even then, it's a limited amount of assistance.
There is no other real solution to this problem, it is simply "part of the game" as they say.