Help me make this rabbet cut in the top edge of a cabinet wall
I'd start by making the job easier - Give your router a wider flat surface to slide on, not just the top of the cabinet sides.  Screw two sacrificial pieces of scrap on either side such the screw holes won't be seen.
Perhaps lay a thin flat piece of smooth scrap on top, with a slot in that top piece so your cutter can reach the work.
Lastly, screw or clamp two or more guides on the sides of your top-scrap piece, to act as a fence for the router.
You can cut some amount per pass, and extend the bit then repeat. Slow and steady does this better.

If space doesn't allow that, you could essentially do the same from the side, but then you're fighting gravity too.  Best to use it instead.
As for a cutter, perhaps something more like an endmill would work here.  There is risk of chipping the laminate surface though so you may choose to pre-cut it on the line with a series of strokes of a sharp knife.
Don't be afraid to use a sharp chisel to finish the ends - no round cutter will get in there perfectly.