I haven’t been stopped since I got my CHL (FL) but if I was I would simply hand both cards to the officer. And we do not have a duty to inform.
Michigan has a duty to inform. I was stopped recently by a very professional Detroit officer. Apparently he clocked me a few miles over. I gave him my DL and CCW cards. He asked if I had a weapon. I told him no. He then asked why I had a commercial drivers license. I told him that I drive kids to a Christian camp in buses. He handed them back and said to drive slower.
Ahhh, two get out of jail free cards.