I haven't come up with anything like that to try yet. *A clothes
hanger just bends and isn't long enough anyway. *The fish tape at the
nearby hardware is $45 and I'd rather not spring for that much for
this one-time job.

It should not cost that much. A 25 foot fishtape at Lowes is like $15.

A clothes hanger will not work. Even if it was long enough, its not
flat and rigid to poke through the insulation. It sounds like somebody
really stuffed insulation in that wall.

Just curious, is the electrical outlet already on the wall behind the
TV? If its an an old work box, you can take out and possibly use that
as an access point to snake the wires up the wall.