For what I have studied till know, くて is 1- the te-form for negative verbs or i-adjectives and it's used to conect the verb or adjective to another word. 2- at the end of a sentence, in the case of the て form, is the short version of てください. So I guess it could be that, but I could be horribly wrong about this.
いや。わたしというか父親が警察らしくてさ
