At what point your manager will find out that you have applied or interviewing for a position in different department within kaiser.
He won't, unless you or someone else tells him. Kaiser is a small world - paths cross all the time. There could be a good chance your new manager will know your current one.
Unless you tell him, only after the transfer process is initiated manager will come to know and also there is a possibility that your new manger or your interviewer reach out to old manager.