How do I legallly install winglets on a standard airworthiness certificate?

I don't recall a 337 as changing them to experimental...

337 has nothing to do with EXP/Std reg changeover. It is the number of
the form that stands for "major repair or alteration" and is typically
accompanied by a new weight and balance when major repairs
(reattaching a tailboom for instance) or alterations (like winglets),
although may not be necessarily in every case.

My SZD59 is EXP reg, but I still went the 337 route when I modded my
stock mechanical brakes to a hydraulic Cleveland setup. Didn't need
the w&b for this job, since the part total was within 1.5lbs of the
old setup and that is all predominantly on CofG, but I didn't trust my
old one and wanted to put some weight in the tail, so a fresh w&b was
useful to me to determine how much...

And why exactly do you want to keep the std reg? Do you use it in a
commercial capacity? (obviously not since it has 1 seat...) Exp rating
allows you a lot more leeway with *legally getting your ship/cockpit
dialed in etc. Work that falls somewhere between changing a tire and
needing a 337 still needs to be signed off by an A&P, but a 337 needs
an A&I and is much more involved. Std means anything beyond changing a
tire/simple maintenance level work has to be farmed out to an A&P,
regardless of your skills.

-Paul