How to solve this classification problem: multi-class or multi-label?
So, in regards to your two questions:

In terms of including the screening status as a label: this really depends on what exactly you want your model to do and the applications of this model. If you want this model to help clinicians understand which patients should be screened and which shouldn’t, then of course include this feature as an additional predictor.

In terms of screening status prediction task being multi-label or multi-class: in this case the classes appear to be mutually exclusive. Therefore, this would be a multi-class classification problem. A multi labelling problem is for example predicting which combination of symptoms would appear if a patient experienced a certain disease.