After mulling this over last night, I think this is what I'm going to go with, and I'd appreciate any feedback -
One mech down and with the others seriously hurting, the players will have to divert to scavenge parts - the Rio Grande will direct them to a different crashed dropship, around which an intense battle took place - LSA, Sovereignty, and Union casualties are numerous.
A dozen LSA survivors are all that's left, with a few captured Vestan Sovereignty troopers - I'm going to use this to characterize the LSA and Sovereignty a bit more, since there hasn't been a lot of opportunity for RP at this point, and the players are viewing the Sovereignty in the same way as the bad guys from Top Gun at the moment.
The troopers are in dire need of medical supplies, and know of a nearby hospital that's been seized by Sovereignty forces - they will ask the PC's to retrieve some, and in exchange they'll assist in the rebuilding of one of the wrecked Everest chassis from the dropship as well as additional repairs on the surviving mechs.
I'll run the raid on the hospital as an on-foot skill challenge, as they are trying not to attract enemy armored chassis to this location by using mechs, and the hospital interior is too tight for size 1 frames. If they screw up, I'll have them still retrieve the supplies, but they'll be followed back to the others by some Vestan troopers, likely leading to more LSA injured or dead - at this point they'll be developing a reputation for bringing bad luck to those around them, that will likely follow them to FOB Saber.
After the repairs are done, the troopers will part ways with the PCs, either hoping they'll meet them again at FOB Saber or never see them again, depending on how things go.