What the voice of White Light has is extreme distortion. The telephone voice on Humility for example is a filter followed by compression to the point it is distorted, then slap back delay and some stereo widening, that's why it sounds breathy but that doesn't mean it was intentionally recorded that way, they probably used a dynamic mic like a sm7b or a sm58.
Breathy vocals are extremely difficult to mix for me, however a lot of compression on a rather normal voice can bring all the breath from the background. What I think you can't find it the exact cut points of the filters, start by using the HPF of EQ3 at 24db/oct and cut all the lows until it sounds good to you, then use the LPF and cut the highs until it sounds like Gorillaz, then compress to accentuate the respiration and add distortion.
If you have access to Lindell Audio 6X-500 preamp watch this video and apply that to your vocal track:
https://youtu.be/aBv1in9g4G0