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 non-conducting beam with a positive charged slug constrained to move along it's length. In this case you can construct a free-body 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.
A version of this experiment has been done in real life - famously, by Millikan - using oil droplets in free-fall instead of a charged slug on a beam.