Why would I want to define a modulation index for each tone (DSB-FC)?
I'm no expert on this, but until the big guys give an answer, here's my take:

For $f(t) = \cos(100t)+2\cos(2\cdot 100t)$, so $\omega_m = 100\ \rm Hz$ for example:



The amplitude's peak value$\ =3$ exceeds $A=2$, so there is overmodulation. This answer only addresses your last point of "I'm pretty sure there's no overmodulation".