My homebrew is that bows will count as small arms for the purposes of operative attacks and Weapon Specialization.
Why? Since the level 1 bow has twice the range of the level 1 pistol, it makes Special Weapon Proficiency (Bows) roughly on par with the Far Shot feat. The Far Shot feat applies to all weapons, true, but the Bow can load load grenades so I think they're roughly on the same page. If you required a separate Weapon Specialization, then someone who wants to use a bow is at a significant feat sink for very little payoff.
As for higher-level bows, you can follow the "Projectile Small Arms Track" and double the range. I would also give bows the Analog special trait.
The level 1 Bow the same damage as the tactical semi-auto pistol, twice the range, quick reload, costs almost the same (5 credits less), and requires a special feat to use.
Here would be the progression:
Tactical Bow: Level 1, 260 credits, 1d6 P, 60 ft, 1 Capacity, Usage 1, Bulk L, Quick reload
Advanced Bow: Level 7, 5500 credits, 2d6 P, 120 ft, 1 Capacity, Usage 1, Bulk L, Quick reload
Elite Bow: Level 10, 18200 credits, 3d6 P, 120 ft, 1 Capacity, Usage 1, Bulk L, Quick reload
Paragon Bow: Level 13, 45200 credits, 4d6 P, 120 ft, 1 Capacity, USage 1, Bulk L, Quick reload
Tactical Gybrobow: Level 15, 91500 credits, 3d12 P, 160 ft, Knockdown Critical, 1 Capacity, Usage 1, Bulk L, Quick reload
Advanced Gybrobow: Level 17, 212700 credits, 4d12 P, 160 ft, Knockdown Critical, 1 Capacity, Usage 1, Bulk L, Quick reload
Tactical Gybrobow: Level 20, 715800 credits, 5d12 P, 160 ft, Knockdown Critical, 1 Capacity, Usage 1, Bulk L, Quick reload
I think Hanzo from Overwatch and Hawkeye from the Avengers have proven bows are popular for sci-fi settings. A bow's advantages should be its silence and choice of payload, but it should require special training.
I think these homebrew rules follow that pretty closely.