First note - some answers might assume an AC string; AC current can be measured with current clamps (which essentially makes a tiny transformer). Doesn't work for DC, tho. In general, you will need to put something in line to measure DC current.
What I'd do first is use my old analog meter. It's a very old Radio Shack that was once probably $20. I've found it to work more reliably on circuits (current or voltage) with a lot of switching noise. If you don't have an analog meter, I recommend getting one; any time your digital meter reading seems a bit wacky or unstable (varying over time, or when measuring the same thing in different ranges), pull out the analog for a sanity check.
(If I had an oscilliscope I'd put that across the meter leads next and look at the waveform, but we'll leave that aside for now).
The smart strings have a highly variable current when doing PWM (Pulse Width Modulation). It's full on for a tiny fraction of a second and full off for another fraction, rinse and repeat often. That can drive a DMM to distraction.
It might measure better when at 100% brightness. You'd have to look at it with as scope to see if it really stabilizes to full on or if there is any off time. Try it.
If you really want to measure it with a DMM, you might be able to put an appropriate capacitor across the DMM leads to filter out the variations. Basically there is a small resistor (called a shunt if anybody cares) between the leads inside the DMM (and the analog meter) when measuring current. As current flows through the resistor it develops a voltage which is measured and scaled by the meter. A capacitor between the leads would smooth out the curve. I don't know the right value - it depends on the frequency of the PWM in the pixels, the amp range of the meter (which changes the shunt resistor used), and the current draw.
I would not be surprised if a really good DMM can handle all this in stride; I've never had a really good professional DMM.
