This is one of the reasons I no longer use lights with integrated rechargeable batteries. Instead, I use lights, front and rear, that can use removable primary or rechargeable batteries. They are getting more difficult to find, but they are much more versatile if you ride in the cold or need for a light to longer than a few hours (and don't want a battery pack with cord). My front light is a Fenix BC30 (in warmer weather I use 18650s, and in cold weather I use the same and carry extras or CR123A primaries), my rear is a PDW Radbot (rechargeable AAAs in warmer weather, extra off the same when it's colder or lithium primaries).
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Don't complain about the weather and cower in fear. It's all good weather. Just different.