With the ubiquitous advancement in smart medical devices and systems, the potential of Remote Patient Monitoring (RPM) network is evolving in modern healthcare systems. The medical professionals (doctors, nurses, or medical experts) can access vitals and sensitive physiological information about the patients and provide proper treatment to improve the quality of life through the RPM network. However, the wireless nature of communication in the RPM network makes it challenging to design an efficient mechanism for secure communication. Many authentication schemes have been proposed in recent years to ensure the security of the RPM network. Pseudonym, digital signature, and Authenticated Key Exchange (AKE) protocols are used for the Internet of Medical Things (IoMT) to develop secure authorization and privacy-preserving communication. However, traditional authentication protocols face overhead challenges due to maintaining a large set of key-pairs or pseudonyms results on the hospital cloud server. In this research work, we identify this research gap and propose a novel secure and efficient privacy-preserving authentication scheme using cuckoo filters for the RPM network. The use of cuckoo filters in our proposed scheme provides an efficient way for mutual anonymous authentication and a secret shared key establishment process between medical professionals and patients. Moreover, we identify the misbehaving sensor nodes using a correlation-based anomaly detection model to establish secure communication. The security analysis and formal security validation using SPAN and AVISPA tools show the robustness of our proposed scheme against message modification attacks, replay attacks, and man-in-the-middle attacks.