I would use a router. Within reason, the bigger the bit, the better. I'd say 5/16-inch, plus/minus 1/16-inch.
Within reason, the diameter of the router bit won't matter much, except that a thinner bit:
- Removes less material per revolution.
- Tends to bend more and could easily break (especially at 2-inch length/depth) so the cut has to be made slower.
Because of that, the thinner the bit, the longer it will take to make all the cuts, and the more the cutting speed will effect the accuracy of the cut.
Depending on the hardness of the wood, and the various cutting factors (bit hardness, sharpness, and diameter, router spindle speed, cutting feed-speed, cutting depth), you may need to make several cuts, starting with a shallow cut then increasing the cut-depth until you reach the final depth.
The larger radius created by the larger diameter bit is not relevant, because the radius material will be (mostly) cut away when the router is turned 90-degrees to cut from the other face.
For example, when you make the cut (basically a square-U) with the base of the router against the end of the 4x4, it will leave a radius (inside) for the full 2-inch depth at each of the 2 corners (turning points) of the "U". If you use a larger diameter router bit, this radius will be larger. But, the bottom (the deepest part) of the cut will be square, with no radius along the full cut.
Then, when you turn the router 90-degrees, to make the same (square-U) cut with the base of the router against the the side of the 4x4, the cut at the bottom will remove the radius that was left along the depth of the first cut.
The only thing that will remain is, at the bottom, at the 2 spots (inside corners) where the 3 cut faces meet, there will be some material that needs to be removed. You could handle that in (at least) 4 ways:
- Use a chisel to clean that out (should be relatively easy).
- Switch to a smaller (thinner) router bit to re-cut only the 2 inside corners, then follow up with a chisel to remove the rest.
- Use a drill bit, a little larger in diameter than the router bit, and drill into the 2 corners at about 45-degrees, to a depth of about, or a little more than the diameter of the router bit. This will create a relief "pocket" at the 2 corners to allow room for the corners of the 2x2.
- Ignore it. Just knock off the 2 corners of the 2x2 that will fit inside the cut. This part of the 2x2 is not visible once it is assembled, so the cutting away of the corners does not have to be very precise at all. Just cut enough so the 2x2 fits all the way into the cutout in the 4x4. If you cut a little too much, or even a good bit too much, it won't matter because those corners of the 2x2 are not visible in the final assembly.
I'd say the best choices are #3 or #4. Some may say they're a "hack", but they're probably the easier and quicker ones and perhaps less error prone. I'd probably go with #3.
This could be done with a router, by marking or scribing the cut lines, then cutting "freehand", but if the accuracy and finish of the final fit are very important to you, the use of a table mounted router with fences, or well constructed custom jigs/fixtures are recommended. Also, it would probably be helpful to do a trial run with scrap lumber first, to familiarize yourself with the process. And as always, measure twice, cut once.