That's not the bit that is weird. These are the bits that are weird:

OP doesn't make it sound like the bag was gate checked - "When I got on the flight the air crew said my cabin bag could not fit into the overhead bin" - No description of what did actually happen to the bag at that point.
Why did OP file a lost bag report with some random mailing company and not the airline?
How did this random company get their hands on a lost bag that should be in the normal airline systems as a passenger bag? This company claims to help mail your items back from Lost and Found - That is for items lost at the airport, not bags that were already on a plane.
Why was OP seemingly OK with paying almost $450 to get a bag back which should be free?
Why would this company re-imburse them for any cash at all, given there's no proof it was in there when they received the bag, to send it back to OP.
Why would they then write a cheque for less than the amount that was lost?

Am I on my own in this thinking? Perhaps I am. None of this stacks up to me at all!