Since he is very small, you probably could take him to a large animal vet and have him put down..
More costly, but would be much easier for you.
People lamb at various times of the year because some want the better weather, others that sell meat lambs want them early to sell for certain markets ( easter, religous holidays etc.)
If timing doesn't matter to you, wait until it's warmer.
As for shearing, we had a couple of lambs that wouldn't go to their Mom once they got sheared, they didn't recognize mom.
After lots of yelling, they finally hooked up. But was a different experience.
Even if you got your ewes some jackets ( for sheep) to keep them warm, it's worth shearing before lambing.
Only for a few ewes though. Otherwise a fairly warm barn.
Sorry about your lamb. Hope things work out for the best.