On softer snow, microspike will be able to do anything that crampons will do, you make just have to kick your steps into the snow a little more. i took microspikes and used them with INOV-8s to climb Gannett peak a year and a half ago. they worked great, with a mountaineering axe.
on icier snow, where you really need traction, you will want real crampons, which the Katoola KTS's will be the lightest full steel crampon you can find. the steel ones are much beefier than the aluminum ones, with longer, sharper points. i have a set and i love them.
Personally, i would worry about my feet getting cold in lightweight flexible running shoes if i was doing any extended amount of climbing on snow. i know my feet were a bit cold, and i was climbing in the sun all day in july on gannett. Your shoes will get wet from the snow. wear some vapor barrier socks on the outside of your regular socks, and you might not have to deal with getting cold, wet feet.