How should I replace a cantilevered rim joist?
I can think of a couple ways you might go here. The rim joist doesn't carry much actual load. Most of it is on the common joists. Really, you can remove it and replace it without doing much else. Keep an eye on things as you do so and watch for any twisting movement in the in the joists.

Another approach would be to rip cut the joist halfway down and replace it with two boards. Remove the top half and replace it with a 2x6, then remove the bottom half and replace it with another 2x6 ripped to the correct dimension.