So I got a cheapo 15-dollar "strop" from an Amazon seller, and when I received it, it was nowhere near what it claimed to be.
Primarily, the "leather" was junk.
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So, thankfully I have a buddy who's a recreational leatherworker and had some big scraps. (Just too small for what he needed for his stuff.) He took care of me.

I disassembled the hanging piece, shredded the old pleather or whatever toxic crap it was actually made of, and he cut me a new strip to match. I took the handle off of the old piece and used an awl he had to sew it onto the new leather, punched a hole in it for the screw in the head unit(who knew a paper punch would be perfect?), and voila! He also burned my initials into it using a wood-burning tool.
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Works like a champ.

Wish I had thought to take pictures during the process!