Any item that requires one raw resource (Like a tree; buckets, baskets, etc.) should only give LP up to a point. I believe it was somewhat like this in world 2 where you could only pick sticks until you got ~700 LP.
I love this: it wouldn't be a hassle for the normal case where you need a bucket (who doesn't have at least a few string laying around all the time?) but it would make it much much harder to bucket grind - esp since bucket grinders probably have lower P*E anyway, and thus it'd be that much harder for them to find any string in the first place.
I don't think branches would be enough of a cost/disincentive, but string is perfect.
Or just make the LP gain really low. I don't want to have to waste my time looking for strings when I need buckets for milking cows just cause loftar and jorb decided they wanted to make them the best LP source in the game and they decided that harder to make was the right fix.
Or, as zampfeo suggested, make the LP gain stop after the char has some total LP amount. I like the "require string" idea a bit better, but both would work. The LP gain cutoff would probably be easier to implement...