The Lukmanier certainly may be the best option. I cycled over it about 10 days ago (late February), and it had quite a bit of snow cover. The authorities keep the road open (except after big storms), but it appears that they don't bother to totally clear it of all snow. On the south side, it's not cleared above Campra, which is at about 1400 metres altitude; the next 3-4 km after that are on the shady south side of the valley, so it was covered in hard-packed snow when I was there, despite there being no new snowfall in the previous couple of days. I took it slow, used a really low gear, and pedaled as smoothly as possible; I barely slipped at all despite only have 30mm-wide tires (the other 370km of my 3-day tour were on snow-free tarmac, so I didn't regret my tire choice). After that section, the sun gets onto the road a lot more, so it was mostly clear, although there were occasional, small patches of snow and ice, but nothing extensive. The north side of the pass was almost completely clear, but I took it real snow until I got below about 1500m and then I could be more certain that I wouldn't meet any surprises, so enjoyed the descent from then on.
Another alternative would be the Jullier pass towards St Moritz, then over the Maloja pass down to Lake Como. The Jullier is quite a bit higher than the Lukmanier, almost 2300m instead of just under 2000m, but it is more used and so should be clearer of snow. The TCS website certainly lists the Jullier as being clear most of the times I look at it, whereas the Lukmanier is normally listed as having snow on the road.
The third option would be the Simplon pass, but that would make the distance a lot further from Zurich. However, that is only just over 2,000m, and the road was completely clear of snow when I went over it two days before doing the Lukmanier. A fair amount of heavy truck traffic uses that route though, making it somewhat less enjoyable.
In Switzerland you can use the national cycle routes, links to which have been given by others. They use combinations of quiet roads and a few sections of light gravel trails. However, they are not always so direct, and can sometimes be challenging to stay on without missing any of the frequent turns. There are many minor roads in Switzerland that are also a pleasure to cycle on, but are more direct and often easier to follow than the official cycle routes IME.
Good luck in Italy: cycling there can vary between being a nightmare and being in heaven; hopefully you'll find more of the latter than the former.