Having had both some training in stick fighting as well as a hip injury since I was 16, I have been carrying canes into prisons, courthouses, airplanes, police stations and different countries for decades. I have a doctor's note for it, too - but I've never needed to show it. Of course I never carry a cane that might garner a hairy eyeball on these excursions, because I'm not an idiot.
Now, I own and carry and have carried-on and carried into different countries (and jails and prisons and police stations and secure military bases) a Canemaster Gentleman's cane that has some fighting features built in, but they're not over the top and appear decorative, not aggressive. I've never had an issue with it in these security situations. I have some other canes that would cause a raised eyebrow, and I don't mess around with them - I'm not looking for an argument; I have other business to attend.
When I have carried-on a cane lately, it's been with a collapsible aluminum cane which easily attaches to my carry-on instead of clunking around (and possibly getting forgotten) in the overhead - still a pretty good (fast!) stick given the space in which I'd be using it.
I'd be a
little surprised if anyone at the airport made a fuss about the cane you describe here in the USA, although I don't have a crystal ball - but it could happen (as others have noted). Maybe you can argue your way through - I am certain that I could, but I'm a lawyer and I have a doctor's letter as well as more than a passing acquaintance with the ADA.
Here's my question: if you're
thinking it might be an issue with that particular cane, why are you messing with it?
You didn't ask for advice so I'll spare you: but if you were
my brother asking this question and he needed a cane or simply wanted a stick during the plane ride, I'd advise that he consider something that wouldn't attract a second glance (like the one linked below).
Amazon.com
Just saw your follow-up post: good idea. But if you want a stick, by all means take one - just get one that looks like what a grandma would be issued by a hospital.