I've heard of the old trick of tying the brim sides down with a bandanna, never seen it done.
I have over a dozen wide-brimmed hats of various styles, only a couple have brims that could be bent down like that without damage.
The only time I see that being any kind of issue whatever is if you only own one hat & you find yourself suddenly stuck in the middle of a blizzard somewhere.
Otherwise, just wear another type of hat if it's that cold.

No hat type is equally useful in all temps & all weathers.
My Montana peak/campaign hat is useful in bright sun, ventilated for hot weather, warm enough for temps down to the 30s for most reasonable periods of outdoor exposure, and I like the stiff brim on the "police" version that can shed quite a bit of water without getting floppy. That stiff brim is also perfect for 45 MPH speeds on the ATV where floppier brims drive me nuts.
For colder temperatures and longer exposures I have other "Fudd" types with fold-down ear flaps.
Just use whichever works best for whatever the conditions are.
Denis