OEM fitment on the Chevrolet Tahoe Z71 bumper includes an additional mounting point that secures the brush guard to the bumper. With four mounting points, the whole assembly is extremely firm on the vehicle.
See those holes on the brush guard at the approximate level of the plastic trim in the photo you provided? There should be a small "L" shaped bracket that would allow it to bolt right up to the bumper.
This photograph should illustrate the bracket that I mentioned. It is visible to the right of the fog lamp in the photogaph;
The brush guard will mount securely without the additional mouting points as well though, I don't forsee any issues without it.
Post some more pics, it looks great!