I have a similar problem I suspect, but your initial information is was above my understanding. I also have pulse secure as work VPN and since I have the Hub 6, my VPN works, but certain websites, both internal and external cannot be accessed. There seems to be no order to what I can and cannot access, but is having a massive detrimental effect on my ability to work.
I firmly believe that BT/EE have changed something recently that causes issues with VPN connections.
Several of my work colleagues using BT or EE are having simliar issues.
Did you get any information back from BT on this? Which researchers did you contact as I may try and make contact myself?
Myself and 3 of my work colleagues (on either EE or BT) are still unable to connect to our company VPN despite it working flawlessly for months.
Unfortunately, clicking on the BT link shows that I'm already opted out of this service.
From what I've read, you can achieve the same result by changing your DNS server IPs to something other than your ISP's, such as Google's.
Can you confirm whether this has resolve your issue?