The sidewinder requires that you use the 10mm beeline that it comes with, unless you're mechanically minded and can dismantle it and insert a different rope (also 10mm).
The sidewinder also comes with, I believe, a spliced eye on the leading end, which would, in itself, be impossible to thread into the Zillon. You'd have to feed the tail end through which, again, requires dismantling the sidewinder.
That, in itself, I believe makes the combination of the two strictly unlikely. HOWEVER.
The Zillon uses the same friction 'chain links' as the Zigzag, which is rated/recommended for lines 11.5mm to 13mm. The stock rope/lanyard (which I personally hate...too stiff and bulky) is 13mm. As you go smaller in diameter, the links have to stretch further and further out to grasp the smaller diameter. This puts more and more strain on the moving parts of these links. I don't expect a fail, but I'll go so far as to say that 10mm rope makes the zillon look decidedly unhealthy. In addition, when you release the zillon under tension, smaller diameter ropes tend to "jump", rather than adjust smoothly. This is amplified when you use it in SRT mode (ie with a thimbled prussik, etc.) I'd recommend a rope that fits the Zigzag's diameter specifications.
That being said...I've run Sterling Tendril (11.1mm) on both my Zillon and my Zigzag at various times, and had no issues to speak of. Jared Abrojena at a clinic in Sacramento a few months back had KMIII (11.1mm )on his Zillon. Petzl must've been thrilled...but hey, it worked, though he noted that it slipped just a tad when set up SRT, which I would probably attribute to the stiffness of the rope. Tendril lets it bite just a bit more than the stiff, static KMIII.
I'm running Luna (=Poison Ivy, 11.7mm) on my Zillon right now...had Treestuff splice an eye on a 12' piece and knotted the end myself. Works great; very smooth, and has enough weight to pitch it around branches without being so thin I don't trust it. I wouldn't recommend anything smaller than 11.1mm, because we know, at least, that it works...but that being said, I'd suggest erring on the side of caution and sticking to something closer to 11.5.
Probably more information than you wanted. Hope it helps!
PS. Congrats on your investment. One of the best tree-related purchases I've ever made.