Two tips:

I saw Kate on TheLastHomelyHouse on YouTube in her Hexi finishing folds the hexies to make a straight line without raw edges: https://youtu.be/23AvfHhACs4?si=_KjXiRlaQKpJPU3F

I finished an inherited pieced top by topstitching it onto a border using a zig zag stitch on my machine. This was an insane quilt top so it worked (photo attached).

I am late to the game on this thread, but what I did was made 1/2 hexies and sewed them in to make a straight edge. it's quick and easy since it just a quick short running stitch. Not sure what size your hexagons are, but you don't have to paper piece them - just cut and draw an 1/4 inch seam to follow. I did a whole quilt without papers, just drawing the seam line with an 1/4 inch ruler and pencil. Obviously I like to hand sew but I didn't want to try and bind the edge of an irregular edge.

Since you don't have any idea how old the original fabric is - maybe find an old sheet or pillow case to use. I was surprised to hear that one of the English quilting ladies on YouTube said that years ago she used old pillow cases for the the white in her GFG. Personally I would use white muslin fabric and not worry about it matching. At the edge it won't be noticeable and chances are it will blend in a few years after being in the quilt.

Ditto--awesome!