My core is 3.5" thick. A 4" thick one should fit; like you said you will have to shave off more of the bumper cover. I shaved about 1" thick strip around where my fmic goes, so you'll prolly have to just do a bit more, depending on the mount. I can take some pics of the inside of the bumper cover fitment on mine, or you could just check it out sometime. At 12" tall, you'll have to mount it pretty high to keep it from hanging below the bumper cover. Should fit, though!
There are some pics of my install
here, although I took it off and re-mounted it a bit higher about a week later. For reference, my core is 24x11x3.5.