If the storage area is climate controlled or stays dry, there should be no problem with storing it inside a plastic storage container. I have my Crosby saddleseat bridle from decades ago stored in a cloth bag inside a plastic tub. It’s never mildewed or dried out. For mildew killer or preventative, I use Lysol disinfectant wipes or any of the store brand from Dollar General or Family Dollar. One or two good wipedowns with those has been enough to completely stop mold on any of my problem pieces of tack. It’s never done any damage to color or texture yet. Nothing has dried out or cracked yet, either.
Hanging a bridle stresses the crownpiece and top of the caveson strap and bradoon strap. I take bits off and store the bridles and all other strap goods in containers if they are not being used.
I absolutely WILL NOT ever use Leather CPR on any tack ever again! If you have used that on your bridle, you better wipe it down with Lysol wipes before you store it. Leather CPR is the most fertile mold grower, ever.