Update: Lloyds bank deed poll policy

I posted a few days ago about Lloyds bank refusing my unenrolled deed poll and stating it was "policy" despite them never explicitly saying that on their website. Following the comments, I filed a complaint on their website, and they've just phoned me to discuss it.
The woman kept saying that it needed to be enrolled and that they aren't violating GDPR because on the gov website it says something about them not needing to and that unenrolled deed polls aren't legally binding. I pointed out that the government has accepted my unenrolled deed poll, and I have government issued Id in my new name. She said she'd escalate it to a complaint manager but that they'd just say the same thing. I mentioned that they are knowingly holding inaccurate information about me, but she didn't care.

What is my next move here?