Assuming you're on good terms and you know they'd objectively support you and think it was a good step in your career?

My inclination is not to (mostly so that if I don't get offered the job then no one need know about my failed attempt to leave), but I will need to take some time off for it at a busy and awkward time so I need a good reason. I'll probably just say medical appointment, but then I wonder if my manager will be slightly miffed (if I'm subsequently offered the job) that I wasn't honest with them.

What's the protocol here?