Whenever I read that part of LotR I find it a strange twist that that rope hurt
Gollum this way.
What's challenging about analyzing fantasy
books is until where can one actually search for a rational explanation. Think e.g about the disappearing wolf bodies in Hollin, etc... I could think of the following "explanation", but be well advised not to take this with excessive confidence.
The rope had some "magical" (elven) characteristics:
He cast an end to his master. The darkness seemed to lift from Frodo’s eyes, or else his sight was returning. He could see the grey line as it came dangling down, and he thought it had a faint silver sheen.
‘Then I’m afraid it must have been the knot,’ said Frodo.
Sam shook his head and did not answer. He was passing the rope through his fingers thoughtfully. ‘Have it your own way, Mr. Frodo,’ he said at last, ‘but I think the rope came off itself – when I called.’
It can therefore be assumed that such virtues could not be produced without some use or influence of a
Ring of Power, in casu Nenya, which was kept by
Galadriel in
Lorien where the rope was made.
Now as to Gollum, he was all consumed by
the One Ring, the antipode of all that is good and pure, and so it can be assumed that any confrontation with
Elvish creations would be a torment and an object of hatred induced by his subordination to the will of the One Ring.
You may draw a parallel here to Gollum's disdain for
lembas food.
If this holds, then indeed your suggestion (that his connection to
the Ring is the consequence) seems to be a correct conclusion. Also, Gollum was hobbit-like before he obtained the Ring and the rope does not affect
hobbits at all: this difference cannot be "explained" without the influence of the Ring on Gollum.
If so, then this rope possibly did not affect
orcs the way it did to Gollum. However, orcs were equally abhorred by lembas food:
they (=the orcs) disliked the very look and smell of the lembas, worse than Gollum did. It’s scattered about and some of it is trampled and broken
So that does cast serious doubts on what this rope would do to orcs. It's a long shot ... maybe we should test it!

PS - welcome to the forum,
Grima!