Sorry to be late to the party!
There is a shim on the end of the mainshaft that Honda used to control this issue. Back-cutting sort of solves it by pulling the gears together: the shim on the end pushes the slack out, toward the clutch side. I think I have a bunch of info on this in another post where one of us was having a similar problem, a search about "4th gear" should bring it up?
Essentially, Honda used shims of .5, 1.0, and 1.5mm (or none) to push the stack close enough together to let the dogs engage their min of 3mm. But after about 40k miles or so, 3mm isn't enough. Good ones started at about 5mm deep, and last forever. So, if you fall into this business again, consider messing with the thickness of the end shims on the mainshaft: that's another way of solving it.
Backcutting is also a good one, just tricky (or expensive) to implement! It's MUCH better if you're racing, too. (We used to call it "Pull-cutting" back in the day, for the way it pulls the gears toward each other.)