I'd recommend having at least two pairs, that would give you time to get yesterday's pair washed and dried before you wear them again.
Wild camping means you'll have to get creative to find places to wash your kit. When I do multi-day endurance events, I typically wear my kit right into the shower, hang it up to dry, and it's usually dry enough to wear ( or toss in my bag) by morning.
How much riding you're going to do each day will also make a difference; if you're doing 20-30 miles a day, with lots of stops for sightseeing, or whatever, then regular athletic clothes (including synthetic wicking underwear) would be sufficient. If your day calls for 50-100 miles of hard riding, you'll need proper cycling kit.
Either way, hygiene is key; you want to get out of your kit, and clean and dry as soon as possible. Absent a shower facility, there are washcloth size 'body wipes' usually used for invalid care, that work well if you have to take a 'Tijuana shower'