In this paper, we design and develop a novel robotic bronchoscope for sampling of the distal lung in mechanically-ventilated (MV) patients in critical care units. Despite the high cost and attributable morbidity and mortality of MV patients with pneumonia which approaches 40%, sampling of the distal lung in MV patients suffering from range of lung diseases such as Covid-19 is not standardised, lacks reproducibility and requires expert operators. We propose a robotic bronchoscope that enables repeatable sampling and guidance to distal lung pathologies by overcoming significant challenges that are encountered whilst performing bronchoscopy in MV patients, namely, limited dexterity, large size of the bronchoscope obstructing ventilation, and poor anatomical registration. We have developed a robotic bronchoscope with 7 Degrees of Freedom (DoFs), an outer diameter of 4.5 mm and inner working channel of 2 mm. The prototype is a push/pull actuated continuum robot capable of dexterous manipulation inside the lung and visualisation/sampling of the distal airways. A prototype of the robot is engineered and a mechanics-based model of the robotic bronchoscope is developed. Furthermore, we develop a novel numerical solver that improves the computational efficiency of the model and facilitates the deployment of the robot. Experiments are performed to verify the design and evaluate accuracy and computational cost of the model. Results demonstrate that the model can predict the shape of the robot in <0.011s with a mean error of 1.76 cm, enabling the future deployment of a robotic bronchoscope in MV patients.