Like barry has said before... it's not just ZDDP, but the high temperature high shear (HTHS) rating. ZDDP only matters once the oil shears. Synthetics usually have much higher HTHS ratings, but yes, they can find leaks more often. I just threw some Rotella T-6 full synthetic in a 179k Miata and it has yet to mark its territory.
I use Kendall GT-1 20w50, as it isn't too expensive and the specs are decent.
Don't need a motorcycle specific oil as those are usually marketed to keep wet clutches from slipping. Just about any non "energy conserving" oil works with a wet clutch. I used to run Rotella T in a Kawasaki Vulcan 750.
Now, my main concern in your case would be that you've had the bike for a year, haven't changed the oil, and who knows how long it was before you got it? Sludge will be building up in there, and your oil is surely getting acidic. Especially if you are doing short runs and stop and go riding.
FWIW, apparently the newer service ratings have included tolerances for E85 gasoline... maybe the ethanol blowby does different things to the oil than the good old juice?