car washes are great for "now" but over time they cause more harm than good. Over the 25 years I've been owning vehicles I've seen my fair share of car wash nicks, scratches and other damage. Some experiences that stick out was a tiny shard of metal stuck in a washing drape make a zig zag scratch right down the middle of a mach 1 mustang I had 15 years ago. Chrome pieces tend to get swirl marks and other increased wear after using car washes exclusively. I've had more soap scum left behind than I care to deal with. Broken antennas, water in headlights etc.
And the last straw for me was last year with my power deployable mirrors. I had them rotated "in" for the wash. Before the truck was through the air dryers; before I deployed them back out, a kid worker with drying towels at the wash came up and manually flipped the driver side out. I was honking and waving at him to stop but he didn't notice. He stripped the mirror gears, boogered up the motor. I don't know. I showed all this to the car wash owner but the mirrors worked for the remainder of that week and he didn't want to take any responsibility obviously. Well now, neither mirror works and I'm done with car washes.
It doesn't get 10*F here, but when it is super cold, I literally put on my duck hunting waders and a slicker jacket and still do my own truck washing in the driveway. I realize in 10* weather the water hose is probably frozen, or if you had a thawed hose the water and soap suds would freeze on your truck before a good washing can take place. I don't know what to tell you there. I'd probably skip washing until it was warm enough again.
But to help you out, if you still insist on going into the car washes. Crack the window a couple inches, throw a thin strap around the mirrors, like a ratchet strap strap and pull it tight around the closed mirror with the ends of the strap going inside the window. Close the window up on the strap ends. You've got to make sure the window closes tight enough so water won't get in at the top. But that could possibly hold the mirrors in the closed position and not let them flip out with the forward rotating carwash brushes.