Although I dont use breadwallet, I think there is no way to export the private keys (correct me if wrong). I recommend Coin Pocket which allows you to export your private key in BIP38, protected by the password of your choice. Simple and reliable.
Breadwallet is "deterministic". You get a new address for every transaction. Your recovery phrase is how you export all your existing and future private keys. Just 12 words, even simpler than a bip38 or wif encoded single private key.
I like coinpocket, but it is a "server trusting" wallet. It relies on and trusts servers instead of connecting to the peer-to-peer bitcoin network like breadwallet.