How can I repair damage to the corner of a 4x4 post?
Any repair as you describe will look like a patch and will make the situation much more conspicuous than it is now. The only fix, aside from replacement, would be to wrap the posts with something new. You could box them with 1x6 ripped to 4-1/4&quot;, for example, but obviously that defeats the point of reuse.
I would probably add a few finish nails to secure the loose bit, then run a sander over it and call it a day.
UPDATE: I totally missed that you were painting. In that case, I'd probably run some screws into the loose area (or remove it), then skim with a good wood filler like Rock Hard Water Putty. Sand and prime, then paint.  I would not install a filler block. That's a lot of putzy work for little gain.