Since it seems to be that a defense of the Axanar project is that it is "not for profit", perhaps even pursuing if not issued a 501(c)3 US IRS tax-exempt status (or even a different subcategory of tax exempt 501(a) or the like), it seems to me a lot could be cleared up by just concretely stating the date that the application was made and what the current status of the application is. Either IRS form 1023 or 1024 needs to be filed
https://www.irs.gov/Charities-&-Non-Profits/Have-you-completed-and-signed-the-correct-application?
PBS is a production and distribution network for media. A not for profit theater company has a theater and rents it out and collects donations. I suppose in theory, all questions of Trek IP aside, it could be possible for Axanar to be a not for profit Hollywood arts organization focusing on SF productions.
Even more could be done to clear up any misapprehension that this is a for-profit venture, by publishing selected key bits of the IRS application if the charity-in-the-making feels it would not interfere with any discussions with the IRS. One part to issue could be the "organizing document" of the charity (required by the IRS to be in the application) which identifies the charitable beneficiaries of the organization. Another bit could be Part IV of the 1023 application, a description of past, present, and future activities of the organization, which essentially is the mission statement and planned beneficent activities. The application becomes public upon approval in any case, but these bits would show intent pretty clearly.
If "not for profit" is not meant by Axanar in the sense of running this studio/production company under IRS not for profit status, or if somehow there is a vision that Axanar/Ares will somehow interact with a charity also to be created and transfer assets to it and rent them back, or simply it should be taken to mean "is not dispersing any dividends", then being clear about what the phrase does or does not mean in these regards would also help.
It could also be helpful to indicate how the various fundraiser funds are going to be routed into the not for profit organization. For example, are the donations tax exempt as of any particular date? Are the assets bought with the funds under the control of the not for profit organization?
If Axanar/Ares is instead just a for-profit corporation which does not plan to have any dividends to disperse after it makes the film, then for profit language should be used, ie, the firm is "not profitable". But this does not mean the firm does not accumulate assets as a result of its income. A true not for profit holds its assets as a public trust and cannot have equity investors. The fact that Axanar leadership has talked about seeking future equity investors is confusing in this context.
Of course, none of this addresses the lawsuit pe se, since the lawsuit says the IP created "financial benefit", and for-profit/not-for-profit corporations can both accrue financial benefit by receiving income.
Being a 501c3 and saying this meets CBS/Paramount requirements to "not make any money" or perhaps "don't make a profit" is an interesting idea, if this is where it is going. I don't personally think it could work, though. Imagine PBS pitching a new Star Trek movie during their fundraising, saying they have permission because they won't make a profit. But who knows?