Chris,

That depends on whether the tension member is connected directly to the
column. In my idealisation it isn't. The tension member is connected to
the pin. The pin is also connected to the support and the column. It
seems like you are disagreeing with my idealisation, which is fine, but
the FBD is consistent with the idealisation. Perhaps someone else can
offer an opinion on this?

Direct or indirect, it does not matter. If it is under tension, it is
connected, and must be included.

You might try summing the forces you believe are acting on the beam in
the horizontal direction. If you use the correct support reactions, you
will be in for a surprise. The expected result is of course zero.

Bill