I would echo Malen’s response that Tunnel Mountain and Stony Squaw are the only two easy winter peaks in close proximity to Banff townsite. You might be tempted to try Sulphur Mountain, it has a cable car to the summit ridge, but the actual summit is a couple of difficult kilometers south of the cable car station and in winter requires alpine technique. I am aware of a couple of failed winter attempts.
Whitehorn is within the Lake Louise ski area, and it is possible to take ski lifts to within a 100’ vertically of the actual summit. You do need to be proficient at down-hill skiing to do this as the final ascent is by poma-lift (not by chair-lift) and you will need to do an intermediate ski-run to descend from the summit, as I am sure the ski-run operators will not allow people to descend the ski-runs on foot. Mount Whitehorn is at the far-left of this map
If you are prepared to drive about an hour east of Banff townsite, you will find a number of foothills peaks such as Prairie Mountain, Stony NW3, Deer Ridge (probably easiest), that have far less snow, and are activated many times in the winter.
Please let us know if you would like company on an activation, I am sure we will find a willing companion.