For winter driving I’d suggest trying to down size your wheel and tire combination and having a dedicated set of winter wheels. A 15x5.5 front and 16x7 rear should be a good combination that should still clear the rotors and calipers (as always test fit for yourself).
With snow tires you want to increase your sidewall and reduce the contact patch. The reason for this is to reduce your spring rate (taller side wall) to avoid “shocking” the tires contact patch on slippery surface, and concentrate as much weight as possible on a given area (reduction of contact patch).
Due to the fact that the Elise uses very unique sized tires getting the “perfect” sized snow tire is a little difficult but I think you can find a decent combination:
From research and experience I’d recommend the Blizzak LM-22 series they offer some of the best grip on snow and ice and are rated to 130mph.

Blizzak LM-22 (H speed rated = 130mph)
-If you can find a 15/16 wheel set
Front
185/55HR15 2.4% shorter only slightly wider
Rear
205/55/16 .04 shorter
If you decide to run the stock 16/17 wheels set
Front
205/45HR16 or the taller 205/55HR16 if it will fit in the wheel well
Rear
205/50HR17
Now these are speed rated tires but are not performance tires so you’ll need to be mindful that your grip on the dry will be greatly reduced. Also being that the Elise is mid-engined and rear wheel drive it should have good traction in the snow, but will be limited by its low ride height. When driving the Elise in the snow you’ll have to be careful what you try to drive over as the risk of being high centered is great.