Can I snap the end of my arrowhead to the grid?

The selector tool always works on the bounds of the object. The "scale line widths" button determines whether it uses the bounds including or excluding line attributes like arrowheads.  The path tools all operate on the raw path data ignoring any line attributes applied.

If you therefore want snap to grid to occur at the end of the arrowhead then please select the line with the selector tool and ensure scale line widths is enabled.