Tag Manager will allow you to easily track button clicks, however if they are taken to a 3rd party site to then book an appointment, unless that 3rd party site provides analytics data, you won't know what happens after that. Not much you can do in that case though. You can only track what you can track. And I wouldn't worry about the bounce rate if it is just people clicking on the button to book an appointment. That's what you want to happen, so the bounce rate is pretty meaningless.