Technically speaking, CBS/Paramount could shut you down at any point in this supposition. In practice they would likely use your distribution method, Youtube, against you by filing a claim that would immediately halt monetization, with a likely eventuality that they would assume control of it and monetize it themselves. It happens and there are plenty of Youtubers who are unhappy about it, especially when their videos actually fall within "fair use" guidelines.
Another question directly related to the Axanar thing is, "What if I am actually a professional myself, who has a bunch of professional friends, and want to make a fan film?"
The IP holder has an absolute right to their IP, provided it isn't being used under "fair use" guidelines. If the IP is being used straight-up and not for parody, review, or educational purposes, then you're pretty much hosed if the take a dislike to you.
Bear in mind, YouTube is chock full of Star Trek "episodes" that are blatant IP violations like Star Trek Continues (that's how I watched it). So Paramount has knowingly or unwittingly allows a number of these to slide. They are clearly not reviews, parody or educational purposes.
Your example of the "we're professionals and we have pro friends who will volunteer to help do this for free in Bradd Pitt's basement" being another variation of the kinds of "have we crossed the line past 2 friends shooting a crappy video" for Paramount to care about.
Here's my basic understanding of IP, Trademark and Copyright. If you don't demonstrate effort to enforce it, you lose it.
By allowing Star Trek Continues and some others (Renegades, New Voyages, etc), Paramount/CBS has tacitly given approval to using their IP.
This may be the real legal battle Axanar was gunning for (remember, their leader IS a Lawyer, they do stuff like this). By coming in as "yet another IP infringer", Axanar may be trying to force Star Trek out of Paramount's hands by way of their lack of consistent enforcement on the other projects.
I of course could be wrong. But when the metric for "now Paramount will sue you" is fuzzy and we're left guessing where the line is based on conversations like this, that seems like ripe grounds for an argument to sway a jury.
The darn shame of it is that Axanar itself looks like a really cool project. Outside of the legal wrangling, I suspect the people making it really care about Star Trek and are trying to make a good product. Thus, I take exception to Wheaton calling them bad names. I think it's more complicated than that.