I preferred the Invader II with Sound Wizard Module. The H&K's dirty tones seemed more compressed, which is fine if that's what you like, but even with higher-gain tones, I prefer more open. The Engl's higher gain channels can be really compressed too, but the Sound and High Gain switches add a lot of flexibility, and then the Sound Module brings even more. How the sound module behaves changes too depending on whether you have Sound or High Gain toggled. There's some complexity and learning curve at first, but it didn't seem unmanagable. If you use MIDI, there's a lot of options for saving/recalling different tones. I like to tweak and really fine tune, so the Invader II is right in my wheelhouse.
I didn't end up keeping either, but I occasionally have to resist the temptation to pick up another Invader II.