I'm wondering what other ways there might be to go about this. I was
considering using a router to make a shallow cut through the formica,
then follow it with a circular saw.

Sounds like a pretty good idea, but I'd look into a spiral downcut bit
to reduce chipping. That might require cutting it a little bit long
(1/8"?) first, then trimming with the router to clean up the cut.
Another idea I've heard is to cover the cut with masking tape, and then
cut it upside down with the circular saw (so the teeth are cutting into
the formica, not exiting it). No, don't hold the saw upside down, turn
the workpiece upside down before cutting it. Of course a good blade
with lots of sharp carbide teeth (maybe a plywood or crosscut blade?)
would help.
If it were me, I'd definitely do some practice cuts on the ends I was
planning to cut off.
Good luck,
Andy