The differences in the tire profile itself aren't going to be significant if you change rim width. However changing rim width to wider might help with the issue of the pads not opening enough and touching the tire when changing the wheel.
Just don't put too wide a rim on for that. Sheldon and Continental say about a 19mm wide rim (internal) is the max for a 28mm tire. Just make sure your brake calipers will open far enough to handle wider. Likely they will.