Сan the XProtect Management Client plugin interoperate with the XProtect Smart Client plugin? For example, process a video stream from a camera in the XProtect Management Client and get it already processed in the XProtect Smart Client?

It is possible to share data like properties or similar. You might want to see and explore Property sample –

https://doc.developer.milestonesys.com/html/index.html?base=samples/pluginsamples/property/readme.html&tree=tree_1.html

It is not the best task for Management Client to process a video stream because of performance and scaling issues. Instead consider making a standalone service that can process a video stream, let a Smart Client plugin interface with this service and configure the service in a Management Client plugin.