Rent the power snake. I have one of that type and it works well and is fairly easy to use. Ask for instructions or watch video. Remove the trap and the cable and head will easily pass thru the turns.
Your use of boiling water to locate the blockage was a great idea. However you forgot one thing. When the hot water hits the cold water it will cool immediately.
In other words the point below the 90 that you felt the difference in temperature was the level of the standing cold water in the pipes when you poured in the hot water. The actual blockage is further down the line. Blockages are almost always in horizontal sections of pipe.
From the looks of the piping I think your house is built on a concrete slab. The pipe coming out of the wall is the vent pipe. The drain piping is located where it is and not in the wall, to avoid having it go through the footer of the slab (weakens the footer).
That piping is ABS. Easily cut with just about anything from a bread knife, to stranded wire, to hacksaw, to handsaw, to power saws. I have even heard that you can cut it with dental floss.
If you cut the vertical section pipe you will have to repair with shielded rubber coupling because you won't be able to get the to sections far enough apart enough to use a glue up coupling.
However, the is no reason to cut anything, the snake will go through all the turns including the ell of the trap.
PS
The bladder thing won't work unless you cut into the drain pipe. The water will just go up the vent pipe. The piping under the floor probably runs to the closest bathroom area. How far is that?