If a beis din made a factual error in their ruling and later verbally acknowledged it but still is not interested to change it (we are sure they are bias), may we go to another beis din to get a “permission to go to court”? Or possibly claim the other beis din to a din torah?
This was a Zabla din torah in the USA, they were not professionals.
You are making two serious allegations. Bias and a faulty ruling by the Beis Din even though they know that their ruling is not relevant.
Each one is basis to invalidate a psak in halochoh and secular law.
However, not every factual error is reason to change the psak. It is possible that the psak remains the same regardless of a change in the facts of the case.
Each Beis Din should share with you what your options of appeal are, if any.
In general, halochoh does not allow for an appeals other than by the BD itself that issued the psak. There are, however, exceptions.
Halochoh does allow for liability of the Dayonim for a faulty psak, but that is not practical.
A professional BD will have included in the arbitration agreement indemnity for themselves. It is possible for a BD to have taken out insurance. It is not common to take out insurance as that would raise the BD costs a lot.
In extreme situations there is a possibly of turning to a gadol hador or BD that is universally accepted as the preeminent BD to hear a complaint against the decision of a BD. This is more of a theoretical issue as in practice decisions of BD are final and binding.