"Sole custody" is almost always the starting point on both sides in a custody dispute. If you want to hold your 2A rights up higher than your custody or visitation rights as a father, that's entirely up to you. Regardless, that's the way the mediator appears to see it, and that's the way most courts will see it.
I sincerely doubt the mediator is actually ignoring either you or the law. A mediator's job is to attempt to settle things at the lowest level. If your ex is filing for sole custody on the basis you OC in the home, and you continue to do so, you'll have given her ample substantiation to elevate to the court, where the judgement is usually in favor of the mother, and where issues like firearms are involved, almost invariably in favor of the unarmed parent.
Right or wrong, that's simply the way it is. Question is: Are you willing to work within that framework, or are you going to try and fight a very large, generally anti-gun, anti-father system?
I can't give you legal advice. I can tell you what I'd do in that situation if I were you, as it was just five years ago when I was in precisely that situation (wife from Ca, I was in the military, stationed overseas). I ditched the guns. I hired a good lawyer and paid lots of bucks. As it was, I have good visitation rights, and that's been enough for me to have some serious parent and play time with my son!
This is one situation where "sticking to your guns" may loose you your sons.
There are the quick access small gun safes for a pistol or two for $100-$300 and Walmart used to sell a gun locker for about $100 that can hold a few long guns and several pistols.
The Winchester model I purchased from Wal-Mart prior to my son visiting the first time after I'd bought my first
post-divorce firearm is Winchester Electronic Pistol Safe, model WM-180-7. It retails for just $50, but I got it on sale at $30. It requires both a combination as well as a key to open it. I installed it, and used it. Had objections been raised, I could honestly respond by saying "I keep my firearms securely locked in a safe approved by the State of California for securing firearms."
If you comply with your mediator's recommendation that you "keep [your] gun locked up at all times," you pull the rug out from beneath her arguement concerning sole custody.
At least on that issue.