Have you tried wrapping the cores with masking tape at the floating end, then freehanding to keep the cores from wiggling around as much? That's how I would imagine doing a setup like that, though I've not personally tried it yet.
Squiddoode's got a good tutorial on fused claptons that you can use to learn how to freehand if you don't know already. It takes a bit of getting used to, but the result will be well worth it once you dial it in. This is the video I'm talking about.
When using swivels, I like to take one of my regular screwdrivers (not the small kind like the ones that come in a vape kit, a normal sized screwdriver like the kind you come across every day) and put it through the center of my spool so it spins more freely. When I freehand, I don't need to have the spool in my hand as I wrap my claptons. I can have it on the floor beneath me and it auto-unwinds itself as needed when I wrap.
Another way is to put the floating ends in a vice, between 2 small pieces of wood attached together with wood screws or anything that can hold the ends fast so they don't wiggle around.