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If I understand your issue properly i.e. getting a frayed, split end through something narrow, I deal with this all the time, trying to thread a fuzzy yarn end through the eye of a small needle. This is what I do:

I use an index card for my paper 'guide' - stiffer than regular paper, but not too thick.
Unfortunately, I am dealing with a round eyelet that is only about 0.040” diameter, and the thread (if I don’t squash it trying to measure it) appears to be ~0.020” diameter — before the end frays when cut, that is.
However, your post is quite useful should I need to do some sewing (more likely with a more common material than Kevlar thread!), and it also gave me the idea to more accurately measure these and other small eyelets’ diameters — by finding pins or needles that just barely pass through them. ;-)
Thanks!
Great pic, BTW!
I was going to suggest dipping it in some super glue and forming it for a quick set, but there would be problems getting it on your fingers....maybe use a plastic baggie to protect them when you form it?