You may be happy to know that I just added in a quad_only_ option to the LBT plugin components that generate grids. For example, you can now get a grid similar to the one on the left of your image by setting this option to True.
The method we use is effectively the same as what is in this thesis document and it ensures that all mesh cells occupy the same area.
The component with the new option is currently in the development version of the plugin (which you can get with the “LB Versioner”). This also means that it will be in the stable release this week.