I'm mounting an Outbacker Perth just above a long gutter and downspout.
The gutter/downspout is the ground side of this antenna. This
combination will be fed with 300 ohm twinlead.

The Outbacker or the Hamstick are not usually fed with 300 ohm line. I have
never used either, but it is my understanding they are both designed for mobile
operation and a 50 ohm feed.
If you are planning to transmit with the Outbacker fed with 300 ohm line you
will need a Balanced Antenna Tuner at the shack. Even with this arrangement, I
would not expect very good results because of ground losses and a low radiation
resistance at the antenna input. You will be feeding a very unbalanced antenna
with a balanced transmission line. My suggestion would be to feed the antenna
with 50 ohm coax, and adjust the antenna for the best match at the transmitter.
And yes, you will still need to connect the antenna ground jack to some sort
of ground.
73 Gary N4AST