Will UK companies have to appoint an EU representative in every member state?

No, the provisions of the GDPR provide that once a representative is appointed in one member state that will cover the business for the purposes of all member states in the EU. The only requirement is that the business must hold data in respect of some data subjects in the member state in which it appoints its representative in the first place but after that then that representative can represent the business for all member states in the EU.