IMO, the optimal way to carry my CCW card is in a bifold windowed ID holder I put my driver's license in one window, and my permit to carry in the other window. Almost all my permits are driver's license sized cards.
This way, when asked for my iD, I can hand that holder separate from my wallet. I can place that in my car's sun visor for easy access if I want. It's also easier for me to extract a small ID holder from a shirt pocket versus struggling to remove a bulging wallet while sitting on it and placing my hand next to the gun. Since it's folded in two, I can even wear it over a neck lanyard if I ever need to. Just cut two holes on the crease, and thread the lanyard through.
Most states don't treat a CCW card as a form of ID, so you always have to present a driver's license or other ID along with your CCW credentials.
My method can ease the process of informing the officer I have a firearm. If the officer doesn't ask about me having my firearm after seeing the permit, then I would mention I have a legally concealed firearm and that it's at the 3 o'clock position on my belt.
I see a ton of businesses including Amazon selling CCW wallets that include a "CCW Permit Holder" badge. Lots of debate as to whether or not there's any benefit to such a wallet, but they usually have places for your ID and permit. The selling point is that you can flash it at a Cop while you are engaging a bad guy -- or wear it on your neck -- and the Cop won't shoot you. I think those situations will be rare, and how the Cop handles it will vary. I doubt the fake badge will matter much either way, IMHO.
Concerning the other issue of verifying carry guns, in most states it's against the law for law enforcement to take your legally carried weapon from you unless you are suspected of a crime -- assuming you are in a place that's okay to carry. A government representative disarming you for no lawful reason is unconstitutional. This whole "inspecting firearm to verify it matches the license" issue needs to take that into account.