corp
Martial: Limited by how legitimate this violence is in the eyes of the onlookers (especially the other corps). "illegitimate" violence could make other more eager to attack you and could make their attacks easier to justify.
Diplomacy: Corps are "default entities", even people who dislike corps could be contacted by it and probably few people would reject CD outright.
Stewardship: Corps are reasonably good in gathering the "legitimate income", and CD would be pretty free at deciding where the funds should go (to the security goons, to the brainy research, to erecting some zany building, or as bonus to Porky). That is, unless there would be other shareholders and/or investors, in which case they would want corp to be profitable and would jerk Cass around if there would not be enough money. Probably taking investments from Tom Nook would be less than stellar idea.
Intrigue: Probably unlimited options. My read on setting is that everyone spies on everyone and sabotages everyone. If some really weird sabotage becomes known, probably could make some violence against the group more "legitimate", as with the Martial.
Learning: its expected that corps would sink money on research. Unless Investor Mr. Nook would say "no more than half of the profits could go there", no limits. Could get legitimacy hit if public would learn that CD's corp studies something dangerous to others, like some plague.
Expectations: people involved would expect to get profits for themselves. Would attract characters that would want to get money from working in corp, the more the better. The "internal legitimacy" in the eyes of your own employees would depend on money.
Criminals:
Martial: No limits, except how much retaliation CD would expect to get.
Diplomacy: Would be already "illegitimate" in the eyes of the onlookers. Big difficulties in dealing with "lawful" and "do-good" characters.
Stewardship: Could decide to get any sort of income from anywhere., no morals attached Would not attract any sorts of investors probably. Could take a loan from a loanshark and get beat up by Kiryu later for not paying it back. Anyone could, but for the criminal that would be the only option probably.
Intrigue: Unlimited. Again, the criminals would not have any legitimacy to begin with. The nefarious deeds are even expected from them, and criminals doing crimes probably even attracts less attention than if corp would do the same crime.
Learning: Unlimited. Again, even if the public would learn that the gangster CD researches some new awful "TurboKrokodil" drug, this would not be very newsworthy.
Excpectations: people involved would expect to get profits and opportunity to make profits for themselves by doing crimes (and paying you the boss share). Enforcing some code of honour would be probably not easy. Would attract criminal characters with even more mercenary outlook than corps, would need to filter away unsuitable and dangerous (for CD and other characters) recruits.
Magical sport&culture nonprofit:
Martial: Like corp's, but even more strictly limited in how violence could be dished out. Legitimacy, especially in the eyes of populace, would matter more.
Diplomacy: Should be better with populace but corps would probably think uncharitable things of CD and team. Which could be useful if it would make them underestimate the team. Or not useful. Could make you fly under the radar of some dangerous corp group. Or could make someone refuse to talk to you because your group type is unserious "which shows that you are just a stupid kid".
Stewardship: The org WOULD be expected to put the money where their mouth is. The org WOULD need to fund the sport and/or culture stuff. Good thing, if the org would feel legitimate (that is, if people would accept that the org is useful and not hot air) CD could probably get some donations, on top of more commercial fund-gathering. Downside, people would expect to see the results of the org's work. It would not be impossible for the org to sink more money in the security team, or in some adventuring equipment, but it would require either justification ("we need better security as the gang is threatening our stuff"), real or made up - or juggling the numbers and hiding the secret expenses. Making the deception known would really hurt this org, much more than the usual corp.
Intrigue: Technically unlimited as with the corp, if the secret stuff stays secret. If becomes known, would lead to questions being asked, unless perceived as "legitimate" (as in "the other guy deserved it". A lot of people would "deserve it", but some frustratingly would not).
Learning: Yeah, if it's related to the org's professed goals. Or if the expenses are hidden. Which again could bite CD in the ass.
Expectations: people involved would probably expect less money for their work, this being a non-profit. Would be more important to hide the actual secret stuff from the people not in the known. Even characters informed of Giygas would be less acceptable of "bad actions for the people who do not deserve it", and could refuse CD if she asks something too criminal. Like hurting "civilians". Some people would think that the org, being neither corp nor gang, is stupid, no matter what. Would probably attract more do-gooder members, which also could bring complications. If some character would decide to start some crusade against the unrelated bad guy on their own initiative and expect CD to support that. On other hand, the characters and mooks should be much less likely to betray CD for money than corp or criminals.
On the plus side this org is least likely to "degenerate" into "are we the baddies".
(tl;dr it should be pretty close to corp, but while some diplomatic actions would be easier, some stewardship, martial and intrigue actions would be more difficult and/or complicated, if still doable)
this totally makes a charity seem even more akward, really restricting ability to expand and the like.
Hm, no, I believe it should be easier to expand in terms of membership and income - corps and criminals should underestimate this org and dismiss it as "stupid kids doing unserious stuff". But the conversion rate of membership and funds into hired muscle and weapons would be less good than with other types of orgs. To justify enough security to protect the properties and activities? Yeah, should be doable, though would need some public diplomacy. To justify spending tons of money arming commando team to go after Alien target - would be pretty tricky.