1) Avaya Oceana - at present there are not APIs that would allow a third party to develop their own agent for the services it offers.
2) Avaya IP Office - TAPI
3) Avaya Elite Multichannel - TSAPI/JTAPI/DMCC but the comment about applicability remains since multichannel is a super set of functionality and I don't know what your expectations are relative to teh client's ability to access those services.
4) Avaya Blue (Nortel) - AACC related ones, I dont know them well enough to enumerate which ones, you can start here:
https://www.devconnectprogram.com/site/global/products_resources/avaya_aura_contact_center/releases/7_0/index.gsp
5) Avaya Breeze platform - breeze is not a PBX, so I don't see how Breeze would apply to your ability to use your client with Breeze. Breeze has numerous APIs to do various things with in the context of a SIP application server but since it doesn't provide contact center functionality as a service... I think it doesn't fit. Breeze could muddle with a call on its way to CC Elite, and thus be in a call path to your client, but to me that isnt Breeze supporting your client (without CC Elite in the picture)