As I understand, Dropship repairs are handled by Dropship Crew, specifically crew with a role of "Vessel Crewmember" they Random,
have a tech/vessel skill, which is checked against the repair being authorised.
To trigger the repair select any tech in the repair bay (i don't see vessel techs at all) choose the mode (time x3 / x4 etc) and set them to repair. The skill check and time allocation comes from vessel techs (although I can only use 480mins a day regardless).
Using Against the Bot rules the crew comes with the vessel, from the hanger select dropship and "Hire full complement should give a mix of skills for gunners, pilots and crew.
I have repaired engines, armour reloaded ammo this way so far.