Thats the first time I have heard anyone having issues with an officer 'seeing' something above the visor. I suppose its possbile though. I guess the reality is they could claim they 'saw' something no matter where the registration paperwork is located. Although in my case, the ONLY thing above my visor is a couple slips of paper held to the visor with a rubber band. If he 'mistakenly' sees something that isn't there... I think it would be fairly easy to dispute that one.
You wallet idea is a good one. The drawback for me is that other people occasionally drive my vehicles, so the info needs to be accessible for them as well. I like that idea though, if I were the sole driver, I would consider it.
Wallet works well for me being that my wife and I are the only ones driving the car. And so we each have a copy of both of our registrations in our wallets (well, her purse). One time one of her relatives was staying with us and needed to use a vehicle, so I simply made an additional copy of the registration for him to carry in his wallet (nothing that I know of says you cannot copy your registration and I've been doing so for years to make the second copy as they only send you one).