I posted a very similar question some days before yours here. So far I still did not have any success with my texture.
According to this demo example vtk-js actors seem to have an actor.addTexture(texture) method, that works like vtk’s actor.SetTexture(). The vtk-js Actor reference page does not mention it however.
Some properties of the vtk-js actors are exposed through the dash_vtk.GeometryRepresentation's actor, mapper and propertyattributes but these seem to only be the default values of vtk-js actors, mappers and property.
I am now also investigating saving the regular vtk actor/scene to a file that preserves texture and importing back to the dash_vtk scene to circumvent this issue but so far no success.