You’re correct that patch doesn’t seem to contain any LFO parameter. The modulation is there even when the quick controls are bypassed. This leads me to believe that the modulation is embedded as a non-userprogrammable parameter. If you raise the modulation wheel, you can clearly hear a secondary modulation kick in aswell.
If this is, indeed, the case, there’s not much you can do (apart from using another patch). You could try the “VariAudio” option, but I wouldn’t hold my breath in regard to the results. Maybe Melodyne would work aswell.
However, if you are really serious about your sounds (and can afford it) there are excellent Kontakt libraries out there. You could do far worse than checking out NI’s Session Horns or Session Horns Pro:
Both work with the free Kontakt 5 Player, so you don’t need to factor in the price of Kontakt 5 itself.
I’m sure you’ll get a million other options from other members aswell. Good luck.