Add a new filter effect.

Add a new "Morphology" node, connect it to the "Source Alpha" (The transparancy of the object) and set the mode to "Erode" and set the radius to however thick you want the tint.

Add a "Composite" node and connect the first input to the Morphology node, and the second to the "Source Graphic". Set the mode to "XOR".

This isn't perfect, the tint area will be thicker along diagonals than where it's vertical or horizontal and it assumes you haven't set the alpha of the fill to anything but 1. On the plus side, it's quick to put together once you are familiar with the filter editor, you can copy it to other shapes, and it's non-destructive so you can still edit the shape after applying the effect.

For a nicer looking but less flexible option: duplicate the shape, expand the border of the duplicate to twice the thickness of the tint you want, then convert stroke to path, select both and set clipping path.