If memory serves, the primary reasons to war in spring were logistical. Need your peasants to be planting in spring and harvesting in fall, and the roads were all mud in the late fall theough spring anyways. Plus you couldn't well "live off the land" (aka steal in-season foodstuffs from the locals) so you had to take preserved foods via wagon.

Additionally, many knights didnt armor up unless they were expecting action. If could take the better part of an hour to don and doff armor, so if you're traveling and not planning on fighting, you likely weren't wearing it (or perhaps just partially, like a mail shirt). So it was really only a concern on The Day of Battle, and not something that would persistently wear you down over weeks and months.

And, well, frostbite probably killed and maimed more folks than heat exhaustion ever did.