Off the top of my head, I think you're into a significant welding and machining job. IIRC, the S&W 1911 uses a Wilson beavertail, which has a compound radius that's a good deal smaller than the .250" radius that the Brown beavertail uses. So if you stuck a Brown beavertail up to your frame, you'd have a wide gap between the two parts. To make the conversion, one would TIG weld metal on to the frame tangs of adequate thickness, then re-machine the frame and re-fit the Brown beavertail.
So it's fairly tricky and not cheap.
Maybe a better fix might be to go to your FLG and have him locate the sharp spot that's causing your discomfort and melt it.