Im honestly on the fence over whether or not she’d even want to be a princess.
 
Sure, she’d jump at the chance, but i dont think she’d like it if she spent serious time at it.
 
Trixie really has three main ‘goals’ in my mind. Complete magical power, freedom to use said magical power to fuck with ponies who piss her off, and the respect of all others (and only respect - its pretty clear she dont give two craps about being liked).
 
So princess gets her the magical library, and thus goal one is accomplished. And because she is Trixie, goal two and three wil take care of themselves in her mind so long as she has goal one.
 
So she’s going to do a Twilight and isolate herself in the library. And because shes at least based on narcissists in personality (who are famous for holding insane standards to themselves and taking perfectionism to a fine art - and conversely reacting to even smalls failures with massive piles of self-hatred - theres a reason they tend to get far in life) she’ll stay in there constantly practicing and learning her magic.
 
Any princessly duties are just going to piss her off. She’s not going to be like Luna, caring if people dont like her. Respecting her will be enough. She’ll probably care that people see and know her as a competent princess (again - perfectionism) but that doesnt mean she’s going to be a Princess Celestia or a Twilight.
 
Id guess she’d eventually get irritated by the whole thing and steal every magical book not nailed down, half the treasury and go somewhere she could learn magic in peace.