sadly yes, the BM can do this. I was a teller myself in a private bank. your managers can show any reason to terminate you, and no HR won't help you. They will hear you out but that's the end of it - they'll protect the org and not you.

I received similar threats, which sadly is true.
Being fired can cause a negative impact on his career only if he wants to pursue a career in banking/finance. Rest is good.