What is the right way to measure K on an inductor with three coils?
Is it more accurate to take the K measurement for each winding
independently against the others, or should I just series connect L2
and L3 and measure a single K for L1 vs the series combination of L2
and L3?

Choose one inductor as the reference and measure the others individually with respect to the chosen reference inductor. I say this because when there are more than two coils you cannot have true independence of coupling factors i.e. you can have L1 and L2 coupled at 0.9 AND L2 and L3 coupled at 0.8 but you can't then couple L3 to L1 with an independent coupling value because it has an inherent coupling (via L2) of 0.72 to L1.
So, choose your path a little carefully or it might unravel.

Having just read a bit on-line it seems that some simulators will not like the situation if there are not 3 coupling factors for 3 mutual coils and, additionally, those mutual coupling factors must make numerical sense. So, in the example above with L1, L2 and L3 you might need to add a coupling factor between L1 and L3.