Ideally, you'd use a sliding window FFT, that is...
Sample a screen of data (1.5s); FFT
Cut off the leftmost, say, 0.25s, and add the next 0.25s on the right; FFT again
Repeat for 0.5s, 0.75, 1, 1.25, 1.5... offsets
Note that setting your scope for 1s/div may not yield a buffer size of 10s. Check the horizontal buffer size to be sure.
It's slightly worse than this, really, because scopes don't just chug an FFT, they apply a windowing function first. Which kinda sorta throws away the beginning and end of the buffered range, but not completely. So your buffer size should really be more like 2 or 3s, than 1.5.
Whether this will give you a number that's appropriate for the standard you're testing against, I have no idea. This is more in regards to your "new in DSP" -- get a feel for the tradeoff of time versus frequency, and how much data / time is required for some frequency resolution and all that.
Something could also be said about the sidebands of the harmonics (variations of a given frequency component show up as modulation of that component), but that's tricky, and not usually very apparent from an FFT.
I'd be curious if "1.5s RMS smoothing" is defined elsewhere in the standard, or in related documents. The way it's defined for EMC work -- spectrum analysis -- is quasi-peak, meaning, back in the old days a spec was simply a radio receiver constantly being swept across a frequency range. The detector is simply a diode, resistor and capacitor: when RF is detected, it rises at a certain (attack) rate, and when not, it falls at some other (decay) rate. If attack and decay are equal, it tends to filter out rapid changes in the signal, but if attack is shorter (which it is, for the "quasi-peak" condition), peaks will be emphasized in the readout.
There are algorithms to perform quasi-peak functions on FFT calculations, but what I'm getting at is, I wonder if they would use a similar, relatively simple to measure (in the analog sense) method in this case. Namely, that the "1.5s average" is simply a receiver tuned to 50/60Hz times the 21st harmonic, and with attack/decay rates of 1.5s. Actually implementing that digitally, again, not so fun.
Tim