I did some math on Wolf's Dragoon's Zeta battalion from 3028 a couple days ago. Came to the conclusion for sustained operations, with an hour of combat a day, about 3300 tons of ammunition. Logistics absolutely *destroys* most units in the game. In fact, if I was to play a campaign, I would have absolutely NO units with ANY ammunition whatsoever, and all units would be fusion powered.
So assuming techs don't have to deal with ammo and such, my thoughts on a lance would be:
4 Mechs, 1 squad of technical personnel in 2-3 supply vehicles with parts (not little ones, at least 20+ tons each).
2 Fighters, 1 squad of technical personnel, 2-3 supply vehicles with parts, perhaps more with all the various bombs
12 Vehicles, 1 squad of technical personnel, crew members are all techs as well so less support is required. At least a lance, if not far more for supply vehicles. Also, if its artillery you can double or triple the support units.
Mech and fighter pilots don't have "security", but the vehicle company (at least 30+ personnel) will have at least 1 crewman per tank awake 24/7 "scanning" and monitoring higher command radio frequencies. This is the way the modern militaries do it as well.
If mechs are deployed and must "rest", they'd be provided with security, but most likely they'd return to a FOB or collection area for the night, perhaps company in size, so security could be provided as a vehicle company. Worst case, 1 pilot in each lance will stay up all night (taking turns, perhaps 2 hour shifts) keeping an eye out.
BT never really mentions logistics. I think mech bays don't have "supplies", but each has about 2 bunks for technical personnel. Leopards, Unions, Overlords, and other mech-only dropships have very little cargo, as they carry mechs into combat and don't really support them after.
I would think every Union, or at least Overlord sized vessel carrying troops, has a Mule with supplies nearby, maybe even still in orbit shuttling down supplies as necessary.
The US Military has about 9 non-combat personnel per combat soldier. Other countries have less than that, in some cases less than 1 per combat soldier.
Also, those infantry platoons in all the dropships may not all be carrying weapons, but could be support personnel instead.