Nothing Sundance posted names Carter Page directly as an undercover FBI employee.
It is much more likely that Page cooperated with the FBI investigation into the Russians by allowing an undercover FBI to pose as an analyst at his company.
This would comport with both the narrative Sundance posted as well as the testimony Page himself has given, which is that he was a cooperating witness in this investigation
I tend to agree with your analysis. Bradman calling Page an FBI employee is without any foundation. It sounds exactly like what you described.