No. Even though the class feature gives you proficiency, the 3rd level Weapon Specialization gives you the bonus to weapons you're proficient with as part of your first level in the class. For an example, look at the proficiencies you get as part of gaining the exocortex and the text included. You'd need something like the below (emphasis mine) to gain the specialization bonus.
Your exocortex provides you with enhanced combat ability,
granting you proficiency with heavy armor and longarms. At
3rd level, you gain weapon specialization in longarms just
as if your class granted proficiency. As a move action during
combat, you can designate a foe for your exocortex to track.
As long as that target is in sight, the exocortex feeds you
telemetry, vulnerabilities, and combat tactics, allowing you to
make attacks against that target as if your base attack bonus
from your mechanic levels were equal to your mechanic level.
Designating another target causes you to immediately lose this
bonus against the previous target.
Because the proficiency mod comes from a drone mod, and drones don't automatically gain Weapon Specialization, they didn't add in the bit about gaining specialization, unfortunately.
You can always take Weapon Specialization as a feat though.
