clog
You can't snake through the toilet. If sewer goes out high you will have to deal with sewage draining back on you.
If you can get on the roof, run water down that stack and watch to see if it backs up or gurgles in the toilet.
Not seeing the conditions, but you could pull the toilet to snake out or use the stack to snake out.
You didn't mention the kitchen or laundry, so it may be in the horizontal below the bathroom and should be quite easy to open up the drain.