Not having command rights . . .
For tactical
If you are on the field with the House/Corp forces you are supporting you sort of need a 2nd non-PC player. One person is playing the opposing force, the other plays your 'ally' without communicating to you much . . . no huddling to determine what is the best move or plan. The person would give orders. You might also roll a 1D6 or something . . . 1/2 no further instructions that turn, 3/4/5 you get very simple instructions (move right/retreat behind the hill/fire everything on the Atlas), 6 you get detailed instructions (retrograde manuever, use LRMs to drop mines and energy weapons to hit at range, use flamers to start fires in woods for cover).
Somewhat simpler would be for the player's side, if there are supporting forces present, to all have to move before the House/Corp forces. Say its a merc lance and a House lance . . . even if the merc's side won init, the merc's lance must all move before the House force as a default unless the House commander lets a merc unit move later. To me this may be the most punishing method because I play manuever . . . and its hard to outmanuever an enemy when you are frontloaded. It also means that the enemy will make their move to isolate/damage the merc units- so just like we hear in fiction the merc mechs are out front being used as a damage soak to protect House/Corp equipment.
Forced withdraw conditions could be changed to require more damage or specific damage. You might need to get House/Corp commander's approval to withdraw a damaged unit.
For strategic
You get handed the shit end of the stick . . . the mercs get sent out as the diversion, told to hold out for 10 turns and then House/Corp reinforcement will arrive on map. Or they will strike at the real target once the mercs have diverted attention/garrisons, which will 'give the mercs some relief as the defenders rush back to the target' . . . or not since the House/Corp force delays letting the mercs draw more attention to make it safer for loyal forces.
Your contract says you are to be supplied ammunition by the House/Corp. Sure, the local commander you are attached to gives you access to their supply train and the munitions available- but the supply trucks will carry the House/Corp supplies first and will squeeze yours in as able. Or will let you have the remains of the ammo stocks, their mechs get full loads of LRMs for the LRM5s that two lances of Vindicators, Thunders, Sha Yu and Duan Gung but your force which was assigned as fire support has only half loads for the Archers and Yeoman mechs. Or AC/20 & Gauss Rifle munitions are declared mission critical and can only be issued (or only issued to allied forces) on orders of the theater (3 jumps away) or planetary commander. Or Gauss Rifle munitions are mission critical and can only be fired at medium & short range; or fired when the computer gives a probability of a hit at better than 70% (THN 6 or better).
Could be as simple as being told what your munitions mix might be or might be restricted to . . . such as SRMs not being allowed inferno or LBX not having cluster- 'You job is to hit the mechs, tanks and fortifications.' Or perhaps being ordered to load inferno while House/Corp units do not so if any fires get out of control its the mercs fault- cue bus rolling over bodies. I would say this is something the GM runs.