REI kids has most of what you are looking for. We bought our
hi-tech socks there, we also bought
snow mittens. The gloves were so expensive they are on inventory control of sorts. We don't use them for cycling as a crash would likely result in rips to the glove. Wind/snow pants that aren't too bulky might help some. That could be hard to find though along with a thin balaclava.
Some of the stuff can be pricey but if you know it will be well used, the sting is not as much.
For 15 minutes riding at a stretch, what you have plus some extra bits should work out ok.