I often gently wrap the around a TP cardboard tube and then run a long loop of yard through it - then wear it like a necklace when attaching the binding by machine. Not sure if this would solve your issue, but it works for me. I've also seen people use a similar method but had a wooden 'spool' instead of the TP tube.
Here's another thread I stumbled on while searching for a picture of the wooden spool...
Pix of how I make my "binding spools(buddies)"
Using a pool noodle is a great idea!
Hope you find a solution.
Blessings,
Donna Mc