Assuming you are just dealing with the burns, and not airway issues, the best treatment may actually be in your tent, not in your trauma pack. Dry, clean sheets make great burn dressings. Preventing hypothermia will be key in this setting.
Fortunately, in this given scenario airway burns/compromise is not as likely to be an issue but should be observed.
From my trauma kit, IV fluids and pain medications are the primary things that would be used but I'll have my scalpel handy in case an escharotomy is necessary in the event that any of his burns are circumferential.
^Obviously a professional here.
Additionally, if the burn wounds are severe enough, chances are the person will not survive anyway, outside of a burn unit. Dehydration, shock and infection will end the severely burned person's live relatively quickly (3-7 day, maybe more and it won't be pretty - very painful). Depending upon the severity of the SHTF scenario, euthanasia may be an appropriate option.