Look for the well worn knife in the bathroom somewhere.

Is there a visible plunger or poop knife in the vicinity of the toilet? If so, your decision is simple.

Either way, I’d still choose the toilet. Don’t put too much paper in at once, and if it clogs, just quietly inform the host. Toilets clog all the time, and it’s not a big deal. A year from now, people won’t remember “the dude who clogged the toilet”, but they’d definitely remember “the dude who pooped in the backyard”.

“knowing that you’ll clog it”? WTH? Did the host only provide a piss-only toilet?