Using a shoulder plane makes it pretty easy to get a nice fit. Get it close with the saw and router, then take a stroke or two with the plane, then repeat the exact same number of strokes on the other side. The setting on the blade determines how much you shave off on each stroke and that can be very thin, so it is easy to shave, trial fit, shave until it's just right. See pix.
The japanese saw cuts on the pull stroke and is razor sharp so you cut down to your mark, route the bulk as described, then use the plane. A shoulder plane cuts the full width of the plane. A Japanese plane cuts on the pull stroke, which is easier, but can not cut all the way to the shoulder, you will have to go back and use a chisel for that. Note the stiffener on the back edge of the saw in pix.
Theoretically, you could use a wide chisel to fit the tenon, but it requires very careful work and is much easier to overdo than with a shoulder plane. Just FIY, I added a pix of a wooden Japanese pull type plane.