If I want to automate my VCA to lower my drums channel a couple dB during a verse, because the signal going to the drum bus group channel is lower, it doesn’t trigger the drum bus compression as much, and so the character of the drums overall changes between the verse and the chorus.
How do I maintain steady signal hitting the compressor on the group? Would I be better off automating the group channel rather than the VCA fader?
Put your compressor on a Pre-Fader Insert slot (if not already)?
but if it’s on the group channel, lowering the VCA volume will lower the input signal in the bus compressor, regardless of the pre or post slot.
Wasn’t sure where the VCA or indeed the compressor was from your original description.
If, following your comment above, the VCA is on the individual tracks going into the Group that the Compressor is on then yes, the signal will indeed be reduced before it hits the compressor so the pre-fade insert (I suggested) won’t solve your problem.
If this is the case, then for the Chorus I’d automate the Group Channel itself as you originally suggested.
You could also assign the group track to a VCA if you want to control more than the drums (say the bass) together.
VCA’s can also control other VCA’s, but that can quickly make ones head spinning 
Obviously.