The current system for moving stockpiles over large distances involves a multiplicity of wheelbarrows. If the intent here is to make stockpile transport resource expensive (lots of leather and glue needed for half a dozen barrows) simply jacking up the price of the cart will accomplish the same thing without making the process feel hacky/exploity.
The idea is basically thus: Instead of lifting a full wheelbarrow and depositing it into the cart, the wheelbarrow can interact with the cart on right-click like it does, e.g. the Tar Kiln, dumps the stockpile it's holding into the inventory. For the cart, this takes the whole stockpile regardless of size, and fills an inventory slot. If you want to get fancy, a differently shaped graphical cue can appear than the classic 'grey-mottled tie-bag' but it really doesn't need a different visual cue if that's a PITA; simply returning an error of 'you must unload this with a wheelbarrow' is sufficient. Right-clicking on the bag that represents a stockpile while driving a barrow will result in that barrow loading that stockpile from the cart and proceed as normal.
I would then also suggest that the wheelbarrow's ability to be lifted while full be removed. This helps preserve some of the cost: you'll either give up the 6th slot to move the empty wheelbarrow around with the cart full of stockpiles, or you'll need wheelbarrows at both ends of the transit.
If you'd like to further restrict this, adding some kind of rule that you can only have one kind of stockpile in a cart/boat/wagon/knarr at a time, i.e. no carrying dirt piles AND feather piles, or mixing onions and rocks, etc. This helps limit the hauling with freight systems to single-type bulk freight.
This would be a substantial quality of life improvement without adding additional capability.