Well now, that's ambitious!

No reason you can't do the planned surprise ceremony then follow up with a civil ceremony afterwards to make it "legal." Heck, hit Vegas on the way home and be married by Elvis.

FWIW I know one couple who got engaged on a trek in the high Andes. Even that took some work, including lugging a bottle of wine many miles and many thousands of feet, up.

Good luck!
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--Rick