Hi Coyote,
Most manufacturers work through partner companies rather than direct with customers.
It is true that BT are just another reseller for Avaya (and Cisco). There are lots of other companies that could provide solutions from manufacturers such as Avaya, Cisco, Shoretel etc.
These companies will be all over you to get your business – you are in a powerful position at the moment so make the most of it.
From what you have said about multiple sites it may be that a centralised call processing system such as Cisco CallManager would be a good fit. Basically you put a bunch of servers in your data centre(s) and these control IP enabled phones at your sites. Access to the outside world is via gateways which can support traditional ISDN connections or IP based SIP connections. This solution runs on top of your data network so that has to be up to scratch.
My day job is designing and deploying Cisco communications systems. I would be happy to have a chat with you about them if that would be of use.
Whatever you do though appreciate the strength of your position and do not gift the business to your incumbent without exploring your options.