Ok so here's an update, and maybe some helpful information for someone like me who wants to do this but doesn't know much about the subject.
I started by removing the old bowtie, which includes the opaque yellow-tinted plastic and the inner piece which is covered in a foil-like material. Using a pry tool, such as a putty knife or trim removal tool, the first plastic piece just popped out, really no problem.
The inner piece is held tight with two large plastic pins that go through the grille. The pins are held in place by nuts that in my case took a 7mm socket. The pins are long, so make sure you use an extended socket (don't know the technical name). These nuts are thin and flimsy (don't know the technical name) so once you get them started they can be loosened by hand. To get at them just reach down between the grille and the radiator, right behind the bowtie. Guys with big hands may want to loosen or remove the five 10mm bolts holding the top of the grille in place.
Once those nuts are removed I used my pry tool and my hands to pull the bowtie free from the housing. This required breaking it free from a piece of 3m tape, and required some force. Once this was done I removed as much of the tape as I could, then used an adhesive remover solution to get the small stuff. A 3m eraser wheel would work better and faster, but it runs about $40.00, and the solution works fine, so consider that.
Here's where I ran into my only problem. The new bowtie has two studs that are supposed to line up with the pre-existing holes used by the old pins. In my case, the studs were about a centimeter too narrow, and no amount of pressure got them in place. To fix this, I used a dremmel with a cutting bit to cut away the inside of the exterior holes on both sides, thus widening the holes enough to get the new studs to fit. I then found that I had to cut away the same sections on an interior guide hole in the grille so the new bowtie would fit flat. Once this was done I was able to screw the new nuts on, and I was done.
I'm not sure if anyone else has had this problem, but since I got these from a guy with an '08 Tahoe and mine is an '08 Suburban, I figured someone might find this useful in the future. Once I figured out what I had to do, the whole process took about 10 minutes. Below are a couple of "before" pics, showing the issue. The first is the bare housing, the second is the new bowtie lining up with the housing, showing the difference in allignment. Hope this helps, I'll throw up some pics of the finished product soon.