The administrative (regulatory) complaint should not have any impact on the civil lawsuit except that the contents of your complaint and any statements made by any party about the admin complaint or in the course of the admin investigation are likely admissible evidence in the civil action. It is unlikely that a court would stay a civil proceeding pending the results of an admin complaint and investigation. More likely, a regulatory investigation would be stayed until after a civil action is ended. If the bank has a number of regulatory matters pending (as is sometimes the case), it is unlikely that either the civil action or the admin proceeding will be stayed. There is no real need for a stay of either proceeding.
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I agree with my colleague's answer that the regulatory complaint would not stay the lawsuit. I would also caution you that if you try to have a regulatory complaint initiated and it is not undertaken nor successful, then the bank will likely use that fact against you in the civil lawsuit to show that you unjustly retailiated. While it will not likley impacts the merits of the case, it could adversely impact how you look to a judge or a jury.