This following isn't legal advice.
Do bicyclists travel armed?
Rifles, shotguns, crossbows, spears and swords being rather problematic for cycletouring, I'm assuming you mean here 'handguns'*.
So. Hmm. Offhand I can only think of Thomas Stevens (
Around the World on a Bicycle, Colt .45 revolver) and Dervla Murphy (
Full Tilt, Beretta .32 automatic) who did (well, and admitted it). Both ~in their own words~ were exemplary in when to keep the firearms under wraps, when to brandish it to defuse a threat situation and when & where to fire.
Anyway, the typical USA cycletour is significantly more bucolic and pastoral than Hollywood movies might lead one to believe and I'd really, really seriously move a portable firearm to the unnecessary pile except...
...if one was a past victim of rape or violent assault
and had enough therapy to where they absolutely for sure wouldn't trigger (sorry, but that's what it's called) in a threat situation and they'd trained when to
not use & how to use
and their nagging insecurity prevents them from launching on tour without personal protection, then who am I to say 'don't'.
When I first read the inquiry, my knee jerk response was 'there's no way that would be legal'. Surprisingly,
with the right carry permit, a US citizen could cycletour in most of the USA legally carrying a handgun.
https://www.gunstocarry.com/ccw-reciprocity-map/Transiting in the few states where carry on the person it isn't legal, the firearm could still be legally toted along in the panniers unloaded and locked.
The Kel Tec P32 weights only 280g (less than 10 oz) fully loaded with 8 rounds.
*Interestingly enough, there are some goofy 'knife carry' laws in the USA. Ah, you're goin' have your little utility knife in your panniers anyway so we don't need to worry about that.
Non-US residents: pretty wild, huh?

This preceding isn't legal advice.