Apparently the backpack doesn't recognize Tinker's Construct longsword and mattock as tools. Having the pickaxe in my hand, I shift-scroll to the longsword; then it just scrolls through the hotbar, instead of swapping the longsword to the mattock. Same happened with the mattock. Didn't try any other TC tool yet.
EDIT: I forgot to mention I can't directly put anything in the green slots. I must fill all normal slots and then shift-click from the inventory. If I try to simply put something there, it just doesn't do anything: the object "stays clicked" until I click on another slot.

NOTE: I didn't try any other backpack than the cow one, as of the time of this writing.

Well, the longsword is a weapon, so it shouldn't accept it. The mattock will be added to the list of allowed tools. The code for this uses a fuzzy recognition method to try and accept any tool, but it's a lot of work to try and include every possible tool.

Oh, so weapons are considered not-tools? My bad.
I'll try with some other TC Tools, taking care to not include weapons. Is the scythe considered a weapon?

yes i think it is