Yes, it would be best for any group seeking tax exempt status to obtain a new incorporation so that it would not involve looking back decades. And yes, it would cost a few hundred dollars. With Phase II spending $50,000-$70,000 a year in AFTER TAX dollars, which could represent, if tax deductible, as much as $10,000 to $12,000 in tax reductions per year (I'm only looking at Federal taxes, as NYS taxes would not apply to every donor... for Phase II, in fact, I suspect Federal taxes would not apply to every donor. I'm sure some of the money comes from overseas... or at least just over the border) So, in 5 years we are talking about as much as $50,000 to $60,000 in tax reductions... almost a whole year of donations. ... not doing this is penny wise and pound foolish, (pardon the switch to British currency). I suspect Phase II would find that money could be raised from its higher-income donors just for the rearrangements. And if James then donated the set and other goods controlled by the current Phase II to the NEW entity, the donation would be tax deductible to the donor or donor entities. OR, he could donate the use of the set and charge the entity actual costs of upkeep and rent (or expenses like real estate taxes if he owned the location). The value of the set is not something I could guess, but I'm sure that there are people who can evaluate that. Phase II might even have a high-income lawyer Trek fan who would do the work as a donation. The only way to know is to ask.
In the end, this is not for me to decide, but only for persons currently involved with and working on Phase II.
My sister felt as you do about my father's WWII veterans' pension. I did the paperwork, and in the end about two days work yielded for him on the order of $30,000 in pension payments. (It was our work, not his. He had had dementia for over a decade and had spent down all his savings to the assisted living facility. He was broke and his assisted living facility was going to eject him for inability to pay, as he was in a state where it was legal to collect over a million dollars over a ten year period and then eject a person who had become broke). With the extra pension money, we added money of our own each month and he was able to stay where he was accustomed to living until he died.
Likewise, people may buy opera tickets without a tax deduction, but they are more apt to DONATE to the opera because it's a not-for-profit tax deductible cultural institution.
CBS does not seem to have objected to Farragut's 501(c)(3) status, although I'd be happier of somebody from that group could post here and comment on that matter.
There was a time I did this kind of work but it's been decades and I am no longer familiar with the details involved.