It's not necessary to play a full round for a par 5 eagle as you can simply go to play now mode and set the amount of holes you wish to play on any given course which includes playing only the par 5 holes.
I still require a couple of courses to master myself as I got pretty bored playing every event for 7 full seasons, which is also very time consuming so i skipped a fair few and simply ensured that i was always in the top 100 by the time Augusta came round on the calendar.
I would say that if that is the way you plan to play your career then you should easily master 16 courses over a 7 year career, my only tip would be if you don't have the collectors edition of the game then in the tournaments where you are given the option to change the location then make sure that you change it to some of the courses that aren't available more than once in a season like The Els Club, Kiawah Island, Royal Melbourne and Royal County Down.
Good luck in your quest.
Edit: I forgot to mention that when you do the Tiger Legacy challenges these also go towards the totals on course mastery so it helps to get this out of way early so as to avoid grinding out 100 pars etc when you may not have needed to. Hope this makes sense but if not feel free to message me on PSN where I will attempt to explain myself a little better.