Print Openlayers WMS map with fixed scaling

Question

I need to print a map with a fixed scale (1:500 and 1:1000) on A4. The online map is done with Openlayers, the print request is sent to QGIS Server using WMS getPrint function.

I mange to get an approximate scale of around 1:500 (1:594 horizontally and 1:554 vertically), but I can't figure out how to fix my formulas to get an exact solution. The formula I use to calculate the extent in meters is:

extent_width = width*printResolution/screenScale
extent_height = height*printResolution/screenScale

To calculate printResolution in m/px I use an approach described in this post:

  function printResolution(scale) {
    var unit = map.getView().getProjection().getUnits();
    var inchesPerMetre = 39.3700787;
    var dpi = 120;

    return scale / (ol.proj.Units.METERS_PER_UNIT[unit] * inchesPerMetre * dpi);
  }

To calculate the actual screenScale I use the approach described in this post:

  function screenScale() {
    var unit = map.getView().getProjection().getUnits();
    var resolution = map.getView().getResolution();
    var inchesPerMetre = 39.3700787;
    var dpi = 96;

    return resolution * ol.proj.Units.METERS_PER_UNIT[unit] * inchesPerMetre * dpi;
  }

And finally I put it together calculation the size of the print box on screen using:

  var w = Number(size[0])*printResolution(scale)/screenScale()*18000;
  var h = Number(size[1])*printResolution(scale)/screenScale()*18000;

  var bounds = map.getView().calculateExtent([w,h]);
  var printBox = [
    [bounds[0], bounds[1]],
    [bounds[0], bounds[3]],
    [bounds[2], bounds[3]],
    [bounds[2], bounds[1]],
    [bounds[0], bounds[1]]
  ];

So I'm aware of errors in the calculation (as I have to multiply by 18000 to get an approximate scaling), but I can't figure out what is wrong.

Here is the complete code in a jsfiddle showing the described behavior.

PS: This is a follow up to this question posted some time ago.