Hmmm...good question. On my Laredo 288RL, my outdoor shower is on back wall and the water pump is immediately in front of it inside under a small "end table". On mine I'd probably stuff a piece of faced 3 1/2" fiberglass insulation inside the outside locking door that covers the outdoor shower fixture. Then I would take off the small 10"x10" square panel at the base of the "end table" and leave it open so as to get some of the inside heat into that area.
Another option...could you position a small wattage bulb...maybe 40 watts just inside the wall where the outside shower fixture sits? Again put fiberglass insulation on outside. Heat generated from bulb should keep things above freezing.
I haven't done this...but I'm thinking either option should keep outdoor shower from freezing.
I am sure you will get other suggestions from others here much smarter that me.
2018 Laredo 288RL Travel Trailer
2017 Silverado 2500 Crew Cab LTZ, Z-71, 6.6L Diesel
2017 Bullet Premier 19FBPR (traded)
2011 Silverado Crew Cab, 4wd Z-71, 5.3L (traded)
K9 Handler of my SAR partner and best friend Jeter!