I wonder if those vacuum storage bags would work for your bridles -- take them all apart and give them a good cleaning/oiling then lie all the pieces flat in the bag and suck the air out. You wouldn't have to worry about moisture or mold getting in.
Vaseline the harness and wrap it in an old sheet or two then put it in a plastic tote with a lid. If you have a basement that doesn't get too wet, put the tote there since it'll be more climate controlled.
Be careful with bridle bags: found out the hard way when I decided to store an extra bridle and some strapgoods in a fleece-lined nylon bridle bag in our tack room. I checked on the bridle, opened the bridle bag and it was a solid lump of grey fuzz. I guess it just didn't breathe well enough for our slightly damp tack room and it just molded. I ended up scrubbing everything with diluted bleach, letting it dry all day in the sun then conditioned it generously with Leather Therapy Restore/Conditioner. Lesson learned.
What seems to work best has been a pile of spare strapgoods I keep in a cotton duffel bag in a rubbermaid tub in my horse trailer. They haven't been touched in two years, and there is not a hint of mildew anywhere. Breathability seems to be a big help in tack storage. I have had people recommend Ko-Cho-Line dressing for storage, along with wrapping the tack in towels or newspaper to store it.
Failure is always an option*
*As long as you figure out what you f'ed up and fix it! -Me