a smooth hard object works fine, but in a slingshot, you try to avoid anything that weighs. Means, there is little enough power. Also, hard objects are a serious danger in case the rubber tears.
In your design, I don't think the weight matters, and you have put on a shield for your face anyway. So for your gun, this would work.
Covering the knot in rubber would work, but then the assembly is tough because of the friction.
Here is a picture that shows how rubber typically breaks:
Your photo seems to indicate that melting the cord creates a hard zone which is causing the rupture.
Have you tried not melting the knot ends?
Do you cut off an inch and reuse the rubber?
I finished a 1/4" Type L tubing 24" long piston sealing barrel insert yesterday.
I used a short piece of drilled 3/4" oak dowel, a groove and an O ring.
JB Weld Epoxy used to seal the copper to the oak is curing as I write.
The muzzle attachment is very simple and I can slide the barrel to any position within the 3/4" chamber and lock it. I can 'play' it like a trombone.
Testing should be very interesting.
Your chrono tests inspired me to eventually move the crosspiece forward for 400% elongation. (V=aT)
Member of the gray beard slingshot fraternity.