。
She's been invited to sing in an overseas choir.
Literal
She [topic-は] overseas [of-の] choir [at-で] sing not-want-to-Q [quotative-と] is-being-invited.
Three pieces stack here. The negative-question form ~ないか = '(won't you) V?' — Japanese's standard way of softly inviting someone to do something, often translated as 'how about V-ing?' Wrap that invitation in a quotative と and it becomes the content of someone else's offer. Then make the verb 誘う ('to invite, to ask along') passive — 誘われる — and add ~ている for present resulting state, and you get the polished 'has been invited / is being asked.' The polite ~ています keeps the register neutral. The whole construction is the standard journalistic-or-careful way to relay an invitation someone has received.