So I just ran a round as the TEC Loyalists with the intention of earning the No Exhaust Port Found achievement (don't use any strike craft). As I approached the end of my conquest I realized I had captured (through boarding) a Vasari Transporter that had bombers. "Oh no," I thought, "there goes my achievement!"

Thankfully, I resisted the urge to give in and produce strike craft on all my ">" capital ships that had been begging me the entire game. When the dust settled, I still had the achievement.

TL;DR version: apparently captured carrier type ships with strike craft will not prevent you from earning the No Exhaust Port Found achievement.