PURGE BINARY LOGS taking time
This is a rather nasty bug in MySQL 5.7 that I have experienced outside of RDS 2 months back. There is a strange mutexing behavior involving the act of purging binary logs.
My workaround involved manually deleting some of the binary logs in the OS as listed in bin-log.index and executing PURGE BINARY LOGS BEFORE CURDATE() + INTERVAL 2 HOUR; (Nightly purge every 2AM). It sped up it up some because it had less files in the OS to examine. When it came to the actual binlogs still in existence, running PURGE BINARY LOGS ... simply dragged. That's why I did manual smoke-and-mirrors stuff.
YOUR ACTUAL SITUATION
In your particular instance, you are using RDS. Since you cannot go to the OS, there is nothing else you can do to the binlogs because max_binlog_size cannot be changed in an RDS Parameter Group. You could change the max_binlog_cache_size if you have large or many transactions that need to cached ahead of its writing to the binlogs, but this may or may not help depending on how frequent you write in your app.
RECOMMENDATION
This issue does not exist in MySQL 5.6.
You may have to migrate your DB back to MySQL 5.6.27 RDS.
EPILOGUE
From my experience, I was victimized by this bug in MySQL 5.7.12. You informed me that the bug was addressed in MySQL 5.7.14. I wish I had upgraded like I wanted. Since your situation came up MySQL 5.7.11, I knew someone besides me noticed this problem. Go for your upgrade to 5.7.16 ASAP. Have a good day !!!