I just ran into the exact problem with my 2004 Suburban. Once I got the seat to lay down all the other mechanisms worked as they should and I could get the seat in and out as usuall. I found this post below which explained how to get the seat down. Wish I could take credit for figuring it out on my own but I can't.
"After working on it for a couple of days here is what I did to remove the 3rd row seat
On the rear lower right and left side there are plastic covers, if you push the seat material forward a little you can see a hole where a nut holds it on, remove the nut. On the left side pull back the plastic cover and you can see the wire and the lever that releases the seat. With my foot holding up the #1 release lever on the right side I used an open end wrench
to push forward the left side lever and the seat folded down.
I then used the #2 and #3 levers to remove the seat.
To view the whole cable, open the lower seat cover
, it is an overlapping plastic clip, My problem was a loose cable, similar to a brake cable on a bicycle
, after fixing it I have had no problems."
To remove the nut I used an 8mm socket wrench. My problem was not a loose cable but the #1 mechanism wouldn't pull up all the way, I'm guessing because that left side locked in out of order. I pushed that left side lever down with a hard tool and had my 5yr old daughter pull up on the #1 and the seat folded right down.
Hope this helps if you havn't already gotten it fixed.