He deals with clients about the overdue invoices.
Your answer: あの人は顧客の支払期限が過ぎた請求書に対応する。
My answer 1: 彼は 顧客の支払期限が過ぎた請求書を処理する。
My answer 2: 彼は 支払期限が過ぎた顧客の請求書を処理する。
My answer 2 sounds more natural to me than my answer 1.
Though the position of 顧客 in your answer and my answer 1 may be grammatically correct, my answer 2 sounds more natural to me than the other.
I don't know the true answer of the position of 顧客.
I think "overdue" is more important than "the client" in this phrase, so the more important word is put in the more front position than the other; and the positioning of them make this phrase sound more natural.