I'd say go into your local fishing supplier and pick up some small (size 7 or 10) 'fish rings' they are very small and very, very strong split rings.
Depending on what kind of swivels you've got theres a chance that you may even be able to cut the ring on the swivel and put the split ring directly through the ring on the swivel, cutting out the ring on the swivel entirely, but you'd have to look closely at that.
The size 7 is usually about the same diameter as the rings on the swivel, so its a LOT less bulk.
I'd say modifying the rings on the swivel would damage the integrity of the rig beyond reasonable use unless you were going to weld the ring back together and on that I know absolutely nothing.
I understand you're using them for LEDs but because anyone else can read this cutting the ring would be a very bad idea if you're going to use it for fire.
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