Chris,

Here's what I did. If the diagrams don't match what you intended, let me
know. I assumed that you're going to have a pivot at the top and bottom
to allow the davit to rotate. You may not, but I doubt it will make much
difference because, either way, the anchor points are unlikely to be
highly rigid in a torsional sense.

http://www.mythic-beasts.com/~cdt22/davit_calc.jpg

The calculation suggests that the winch suspension point will be
deflected downwards about 8 inches by a 200 lb load. This is only an
order of magnitude figure, but it is way too much. It means that your
davit would be very bouncy, your top and bottom anchor points could be
damaged, and it might cause failure in an unexpected way.

I fear you are mixing models in a way that does not work. You compute
support reactions based on the ends being fixed horizontally, and then
model the deflection based on a cantelever beam. An FBD of the beam
itself will have (using your top right setup) loads from the bar and
cable (which you might want to reduce to an applied moment).

The compression of the column will move its neutral axis, but I imagine
that the most useful starting point will be to get an angular deflection
at the end of the beam and multiply that by 3 ft to get an estimate of
the load deflection.

Bill