Lets tidy up to make this easier to think about:
You have a conducting beam which is initially balanced and you apply an external uniform electric field horizontally along the length of the beam. This causes a shift in the distribution of electrons to one side of the beam  which would shift the center of mass of the beam away from the pivot point and so it should tip up.
That seems reasonable.
You could imagine a negatively charged pivot and a nonconducting beam with a positive charged slug constrained to move along it's length. In this case you can construct a freebody diagram for the situation ... there would be a force ##qE## from the field, another ##kq^2/x^2## back along the beam, then there is gravity ##mg## acting down.
Have fun.
A version of this experiment has been done in real life  famously, by Millikan  using oil droplets in freefall instead of a charged slug on a beam.
