You can't fight the GM...
The GM is absolutely right when he says that in his campaigns the muleback cords' improvement to a creature's carrying capacity does not apply to carrying other creatures. I mean, really, the description of the muleback cords says
This bonus [to Str from the cords] does not apply to combat, breaking items, or any other Strength-related rolls, [sic] it only contributes to the amount of equipment or material the wearer can carry.
"See how there's no entry for unconscious creature in Equipment?" the GM may say. "You want to transport a creature without a cart? You bull rush, drag, grapple, or reposition that creature; you don't carry that creature."
...But the GM has staked out an unusual position
Except in the case of the cords, actual, Strength-based Carrying Capacity rarely seems to differentiate between what is carried or where.1 Further, for comparison, mounts like the griffon or the mammoth avoid distinctions between creatures, objects, or their loads' locations with their carrying capacities, treating everything as just weight.2 Although some GMs have had difficulty with the muleback cords, that's usually due to an already-high-Strength-score PC hauling around an absurd amount stuff (a bag of holding is cited as a more reasonable solution), not with the PC toting creatures.
The writer earned at most $0.70 for it only contributes to the amount of equipment or material the wearer can carry instead of $0.45 for it only applies when determining the wearer's carrying capacity. Players shouldn't be punished because writers are paid by the word.3
(That unconscious creatures can only be moved with combat maneuvers shouldn't be a thing. There are many situations the game can't or doesn't bother to cover because a human's at the controls.)
This doesn't sound like the real issue
It sounds like you surprised the GM. Some GMs react negatively to surprises, trying desperately to make plots continue despite the interference of pesky PCs. It sounds like the GM tried to stop you with a rules call to further the plot instead of accommodating you and faking the plot. That's okay. That's not, like, the Mark of a Bad GM or something. Plots are hard and GMing is hard; a bad rules call, while not the best way to keep the plot on course, is certainly a way to keep the plot on course.
Talk to the GM before next session. Explain that you're totally comfortable with events as they occurred. Then explain that you weren't trying to cheat when you wanted to carry the unconscious creature but that you bought the cords expecting to have an increased carrying capacity for everything. Then mention gently that, while it's his campaign, other GMs haven't ruled that the cords work the way he ruled, and urge him to reconsider. Finally, if creatures still don't count, ask if you could you get, instead, a heavyload belt.
As a last resort, you can, at the table, lawyer up and explain to the DM, as your character attempts to heft an NPC over his shoulder, that the NPC is food or future material components for the spell animate dead. That's equipment or material, right? Then drop the NPC and say you changed your mind. I recommend against this unless you're planning to leave the game shortly thereafter.
1 That I can't find an instance elsewhere where carrying capacity is differentiated doesn't mean there's not an instance somewhere.
2 Barding is a weird exception.
3 Alternatively, the former phrase might've fit better on the page than the latter. Players also shouldn't be punished for the decisions of the layout department.