I am not too familiar with BlueZ, but I believe BSA would support more profiles.
At least as of now BSA is the one that is supported by Cypress. BlueZ is known to work, but is not not supported by Cypress.
First of all i am not an expert, but I have tried a combination of bluez 5.37 + BCM4339 and have successfully done with A2DP, AVRCP, HFP, Obex. Currenttly able to Stream Audio from mobile and control play pause next using AVRCP. HFP have some issue like audio distortion. Reset of the things are working good with bluez stack.
You can configure A2DP by following the steps in the following link