You only need to send your birth certificate if you do not send your passport - you won't have to provide both of them.
As far as I am aware, you have 3 options:
1) Send your actual British passport (not recommended if you're going to need it for travelling though!), but not your birth certificate
2) Send a certified copy of your British passport, but not your birth certificate
3) Send a normal photocopy of your British passport (not certified), along with your long form birth certificate.
Option 1 requires just the passport, option 3 requires just the birth certificate and option 2 doesn't require either of them.