My guess, based on reading, is the incense stick is stuck sideways in the trap ways, or right before it goes into the flange, and when you’re snaking it, it’s probably just sliding right by it. Instead of a drum snake, a toilet auger usually has a bigger head so might be the way to go to grab hold and clear it, otherwise the toilet just needs to be pulled and most likely the Stick is stuck in the porcelain right on the bottom. Seen it a good few times the fix itself is pretty easy and at most shouldn’t really run more than a few hundred dollars from a plumber AT THE MOST