View attachment 34535 I have done something a little like like this before.
Let the drain sit unused overnight to let the water drain out past the clog if the clog is not total.
I suspect the clog is under the floor where the vertical pipe transitions to horizontal. Here is what I would do:
Get 2 shielded couplings. Let dimension A be maybe 1/4 inch longer than a coupling. Let dimension B be the same length. Then saw out the section of pipe between the middle pink lines.
Do your rodding or drain bladder action down through the space opened by sawing out the section. Then put the two couplers over the drain pipe, and restore the cut out section. Move the couplings to center each on a saw cut. Tighten.
Maybe the better action would be to put a new section in that has a cleanout port. You could do this later if you liked. But my simplified way will give good quick access.