you ride a bike. Is that a bicycle or a motorcycle? Sounds like a bicycle if you don't carry your driver's license.
OR stop being lazy and carry the thing..
You just gave yourself the best advice you're gonna get.
Generally speaking, any activity that requires a government permit or license ALSO requires you to be able to produce said license or permit on demand of a government "agent".
Doesn't matter if its driving a car, carrying a gun, holding a parade or digging a swimming pool, if you can't show the legally required paperwork to the cop /inspector, then you are in violation of something, and a ticket/citation is the LEAST you can hope for.
How likely is the cop to just let you off with a warning? Not very, if there's a gun involved. In fact, there's a good chance both you and your gun are going to be spending time as guests of the police.
Consider the situation from the other side for a moment. You're on a bike (say bicycle so you aren't required to have a license to operate it), You ARE carrying a concealed weapon. You don't have the required permit for the gun ON YOU. You don't have valid ID, ON YOU (the road ID bracelet is information for first responders, NOT a valid ID), and Officer Friendly doesn't know you...personally...
SO, how do you think they're going to react? There is so much that can go so badly wrong, in just seconds, it is a real risk. Some places you are required by law to inform the cop if you are legally armed. Even if its not the law, its considered a very good idea. Some will tell you it better be the first thing the cop hears, "officer I am legally armed, how do you wish to proceed?" can save a lot of trouble. Except, you aren't legally armed, without your permit ON YOU.
So, any level of trust, de-escalation of tension you get from informing the officer you are armed (and following his instructions willingly) GOES AWAY when you can't produce the permit. You just LIED to the cop.
They don't like that...and your credibility as a law abiding citizen is now in the crapper.
You could get responses ranging from "hey, buddy, its ok, we all forget something once in a while, go home, get your permit, keep it with you, have a nice day"..to "ok, sir, you can pick up your gun from the police station when you bring in your permit and show us", or you might be proned out, cuffed and arrested, or all the way up to the worst case, your next of kin filing a wrongful death lawsuit...
Also consider this, the "minor violation" of ILLEGALLY CARRYING a weapon, might be enough to get your permit pulled, possibly revoked.
Carry the damn card. Staple it to your forehead if you have to. If you're carrying concealed, don't leave home without it. If you can conceal a weapon, you can conceal the permit for it, and there is NO valid excuse not to have it with you. Lazy in this case could cost you the permit and your legal right to carry concealed, or possibly worse!
Good Luck!