Removing some of the inner edge is referred to as "skiving". I have done many many tires successfully like this, however I did get a little carried away on a 9.00-16 the other year I was installing on a limber. I had cut into the large cord under the brad, and on inflation it bulged in that spot. End result was I had to scrap a brand new NDT tire.
The best advise I can offer when skiving, is put on a brand new blade on your utility knife. Once you have skived the one side, use another new blade, or turn the blade around on those knives set up to use both ends of the blade.
I actually have an attachment here at work for compressing the tires onto the rims. It is similar to Bob's, but uses a hydraulic cylinder to do the work. It is a must have when doing run flats (one of which I am doing today
).