Shimano 333 hub problem

I built up a lovely kids bike with a Shimano 333 coaster brake 3 speed
hub. The young lad rode it on the ride we took today.

There's been a consistent problem keeping gear #2 adjusted. It acts as
if an internal spring responsible for pulling the hub into 2nd gear is
somehow weak. That is, the little circle with the letter N doesn't
settle down in its target hole. As a kludge, I used a bit of inner tube
as a rubber band on the bellcrank to help pull the hub into 2nd, but
that rubber decayed and I didn't have a spare. In any case, I've
readjusted it several times in the past couple months.

I'd suspect a dragging cable, except the hub shifts into the other gears
just fine. 3rd gear requires the hub to retract the cable even more, and
shows no problems. Of course, I lubricated the cable during
installation. I even spent time grinding away part of the kickstand
mount to give the straightest possible cable routing.

Unfortunately, the bike is an hour away and I normally see it at most
once per week; so some things are difficult to check, other things may
be difficult to remember. But are these hubs extra sensitive to cable
friction? Instead of the full housing, should I consider scrounging
parts to run bare cable over pulleys, etc?

But if anyone has tips, I'd be grateful.

--
- Frank Krygowski