This is your friggin’ EYESIGHT at stake here. What possible reason could you have for risking it by insisting on wearing contact lenses for a hiking trip?
I have horrible eyesight (last checked out at -13 in one eye and -17 in the other), my glasses are nasty looking, and even applying all technology they are still heavy, and slip down my nose when I sweat. Moreover, my visual correction with contacts is superior to what I can get with glasses.
With all of that, I would not dream of inserting and removing lenses repeatedly without access to a properly hygienic environment. I’d just suck it up - and if I wanted to look good in photos, I’d just whip of the glasses while the photographer snapped away, then put them back on.
How many days is this trip, anyway? If you are talking about removing/reinserting one time, and you very thoroughly scrub your hands with alcohol beforehand each time, I could see doing it. But if you are backpacking for a week or so, that is simply too many opportunities for eye infections and you’d have to carry an awful lot of alcohol to be sure you could properly sterilize your hands.
Regarding Bosstone’s idea of washing your hands with the lens liquid, it’s clever but I don’t think it’s sufficiently sanitary. One part of getting your lenses clean with solution is the SOAKING part - the assumption is not that all germs die within a second of contact, but that they succumb after a more lengthy exposure.