Several nations used bike mounted infantry during the war. Finland used it for sure. Sweden had 6 bike regements by 1942 and at least one of the starting units in 36, the one called "Gotlands Militärområde" should be a bike infantry unit, but in game they are a mixed infantry/cavalry unit despite replacing their horses with bikes 1901!
Only complication would be that Swedish army used Skijoring (Swedish Tolkning) for both bike and ski infantry making them more mobile. But I think making it possible to have bike infantry in the first place would be an improvement.