How to store my O/U

Good question, Otto. I don't claim to have all the answers to your question because individual experiences may vary, but I used to leave my favorite O/U in a similar (but slightly more expensive) case to the one you describe and it never hurt it one bit. I would take the gun to the range, shoot the gun, do a quick clean and wiping of the gun, and then put it back in the case and take it home. Then, it stayed in the case in my basement until the next time I went to the range. That's the way I stored that gun for about 3 years and I still have it today. It looks good as new except for some ordinary wear and handling marks from having many thousands of rounds shot through it. I always tried to wait until the gun cooled off for 15 to 20 minutes before wiping it down and putting it away. I didn't want to put a hot gun in a padded case and increase the risk of condensation.