I'm with Will. "Allthread" is not rated for high stresses. Plus it screams "ugly."

If you don't like the look of cable, it might be practical to have hot-rolled steel rod welded or machined to use turnbuckles. I use temporary cables to take up the slack before fine tuning within the limited range the turnbuckle provides. Half inch diameter or larger ought to do it, in my experience.

In restoration work we have spliced in new tie-beam sections with keyed scarfs, using the cables only temporarily. Perhaps you can consider that option, if enough length remains on the original tie beam stubs.

You may indeed be able to reduce sag, but it will be a hard pull unless you provide some vertical jacking force to lift the rafters and ridge while drawing in at the tie-beams.

Your hay-carrier problem is a very common one, by the way, ruined a lot of barns.

Good luck.