That’s a great question, Dory.
Personally, I don’t think you’d have a legal issue (disclaimer: Cheryl is definitely not a lawyer) but I think there are other issues.
At best, the reference will be perceived as being derived from the show. At worst, it could be seen as stereotyping. In your shoes, I’d change the name and make it something far removed from Trixie. So, not Dixie, either.
Also, you used the pronoun “she.” Is this person trans? (There are certainly female-to-male drag performers, but I’m assuming you’re referring to male-to-female drag from what you mentioned above.) Or is the “she” only for when she’s performing? Pronouns are very important to some people in the LGBTQ+ community and you’ll want to be respectful of that.
You might want to ask yourself if this person in your book is a fully realized character or a caricature for comedic relief? Why did she/he/they/zie choose that stage name? This part might not be in your book, but you need to know it as the author.
Also, you might want to think about having a sensitivity reader look over the parts with your drag queen. It couldn’t hurt. I’ve certainly made mistakes based on my own lived experience and inner biases. We all have them.
You are so amazingly good at educating yourself that I know you’ll do your character justice. Maybe you’ve already researched it and don’t need any of this, in which case, ignore me. But I thought I’d bring up some points you might not have considered. I always appreciate it when someone does that for me.