I've decided to go with the aforementioned Conquistador and Colossus, plus two Hercules, a single Vampire II, and a Mule. It means I'm only going with one battalion each of heavy and cavalry armor, but by putting my BA troopers in conventional bunk rooms, the bulk of them fit in the big spheroids, with the Conquistador and Vampire providing maintenance bays. VTOLs for the suits are carried aboard the Hercules, and any spare bunk rooms are used for support personnel or turned into ad-hoc medical wards.
The Vampire is mostly there because I remembered I have a single mini, and the ship can be used to drop teams into areas where a low VTOL insertion would be too risky or impractical. And while not a true assault ship, it can act as an escort for the Ball Pit while the Conquistador and fighter wing do the front-line space fighting.
I tried to find something other than the Mule for bulk cargo needs(I think it's overused), but the Mammoth is too big and even more fragile, and everything else is either too small, or Combine-only.