Enclosures are not that expensive. "Bare" USB to Sata adapter cables for temporarily connecting drives during data transfer are also available.
Maybe I don't fully understand what you are trying to do but ...
Can't you rotate the drives in the external enclosure while doing several incremental transfers of data?
Your computer is also a sort of enclosure. How about mounting one of the drives internally for example by making the new, bigger drive bootable (by putting either an OS Installer, a clean installation of the OS or a complete clone of your start up drive on it.)
You can also make one of the externals permanently bootable to have as a back up from which you can start your computer should the internal drive fail or you can make one temporarily bootable only while you move files around.
Not sure if I described that all that clearly but essentially you have three disks and two positions for seating them in a game of musical drives. Takes a bit of swapping (I'm assuming that's where the word conveniently came in) but you may not have to spend extra money to achieve what you want.
Through my preliminary research, a barebones enclosure would still set me back about $20. That's approximately 20% of the total cost of the replacement drive itself, and for something that will only be used once, I consider that overhead that if possible to avoid, I want to avoid, which is why I asked my question to begin with.
I do not want to make either of these drives bootable. They serve as media storage drives, and I need every last bit of space for the media. I already have a drive dedicated purely for emergency backups anyway, and a 3rd storage device for my critical work/information.
My current internal drive has nowhere near the capacity of my storage drives, so it would take a lot of swapping to get all of the data transferred, which is simply not worth my time (from a pure labor standpoint, I would save money just buying the enclosure at that point). I was just curious if there was a method to directly connect the drives to transfer data without adding to overhead or going through incredibly inconvenient hoops to accomplish my goal, that's all.
EDIT: I wasn't aware of these USB->SATA adaptors, which fit what I'm looking for exactly. They also happen to be cheaper than dedicated enclosures. Thanks for the information.