Thanks, it worked. But omfg, WPF sucks big time so far. Totally unnatural/unlogical compared to what "we are used to" from winforms.
FWIW, if you don't specify a width at all (and no other positioning/sizing properties besides the height and vertical
alignment at the bottom), WPF will make the status bar automatically resize like you'd expect it to. This is the XAML generated by Blend for a window only containing a status bar, and it resizes as expected: