I wouldn't want to do this without indicating on the profile in some way that this is not a profile you are using very often. Maybe a "mark as inactive"?
I've thought about that and I don't think this should come with a public marker on the profile of "inactive" or anything else. Simply, the desire/need to publicly mark a character as inactive is wholly different to the private desire/need to organise one's character picker.
For example I have Princess Bubblegum as a character. I barely use her, and I'll probably icebox her. I would use her, however, if someone was interested and there was a compelling scene we could do. But if she's marked as inactive, nobody will bother messaging her, will they? I'm iceboxing her because it just so happens she hasn't seen activity recently and there isn't likely to be any in the near future, but I don't want to discourage anyone from seeking out that activity or anything!
So I don't want this to also manage the public state of my character, I just want to be able to have a degree of organisation for my character picker. If it marked my characters as inactive, that would be unwelcome and would simply discourage me from using this feature - I probably would use it for one character (a different one) and that's that. Nobody else should be made to care how I organise my character picker, because it has no direct relation to anything anyone other people should care about. It's just me doing my organisation and that's it.
If, like Astarte suggested, we could expand our character picker to see even the iceboxed characters, I'd probably take most characters out of my character picker except the 5-6 I'm actually switching to frequently. The rest already have their roleplays going, and otherwise it's once in a blue moon I need to actually pick one of them in a public channel. The fact these characters aren't by any means inactive goes double in this situation — I just have no need to switch to them 99.9% of the time, so I want them out of the switcher, but the "expand" would be useful for that 0.1% of the time I do want to switch to them. (The more characters I have, the more and more useful Astarte's suggestion will become.)
A feature for marking characters as inactive could be good, and I could see that being applied regularly to characters who're removed from the character picker, but it shouldn't be synonymous with the very action of removing them from the picker.