There is no such thing as "the" Tea Party as a formal political party. It is a term that describes any number of separate groups that refer to themselves as one "tea party" or the other . So, Trump cannot run as the candidate of the "Tea Party" because it doesn't exist.
The fact is, self-described "Tea Partiers" are all over the place when it comes to Trump anyway. Maybe of them like him, and many absolutely despite him. Those tea partiers who consider themselves strong conservatives are much more likely to oppose him.
His only options other than the GOP are to run as a write-in independent, or to bail out early and go for the nomination of one of the pre-existing smaller parties. If Trump is blocked via GOP crap maneuvers then he should run not third party, but TEA PARTY. It’s been a long time coming.
You are legally correct about the Tea party not being a formal party but it would be more about running under a label as opposed to a party. After all isn’t the GOP nothing more than a mere label anymore? It’s become a fools party. What does the old fool Mitch stand for? And then there’s the young fool Ryan. GOP leadership is so stagnant that the people are starving for label that means something, even I it’s single issue such as “build a wall”.