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I would wear it. Wear it working in the garage, mowing the lawn, etc.
It has become my opinion that distressing leather doesn't give an authentic look except in pics and for a very limited time. The wear process is ongoing and distressing attempts to replicate a certain age. Once real wear occurs over the distressed wear, it starts looking funny.
I know many will disagree but that has been my observation.
Now, tumbling it in the dryer on cool with some tennis shoes, storing it outside for a while, etc is much more effective IMO than any abrasives or hand wrinkling. I would always stop before the desired effect and then allow natural wear to bring out the look.
If it is leather, and not synthetic, it shouldn't be an issue. Either dry or wet should work.
I'm not familiar with that brand but I know there were many produced in the 70s and 80s that had a plastic material visible under the bill - almost like vinyl. I purchased one once on the 90s, advertised as original, that I returned as it just didn't look right. I chalked it up to a movie prop or repro from the Hogan's Heroes time period. Every original I have seen has a bill made of leather - called HH by many, it could be a thicker leather too I suppose. Any leather should bend easily enough but if it is synthetic I would heat it up first.
I think I saw a pic of one. One site lists them as an original maker but I'm not too sure about that. Before militaria became collectable there was a muddy area. Now there are examples never seen in the day, like the A-1 visor cap, that appears now as NOS. Things were made from the amazing amount of surplus after the war and many are confused by the age alone.
I would soak it in cool water and mold it how you like. You could sew on some HH strips or accents too. How about a name tag on there somewhere? I'd just have fun with it and make it yours.
Dave