i'm assuming you're talking clincher tires here.
in my experience, continental gp4000s tend to run a little tall for the size (which is why some tout there aerodynamics).
a wider rim will generally make tires sit a bit lower. if you think about it, there is a fix amount of material, and if the beads are stretched further apart (due to the wider rim), then there is less material to extend up above the rim. the difference is likely small, but it may be just what you need.
if your clearance is that tight, i'd give a minute to think about any debris getting caught and potentially damaging the frame and/or locking the wheel.
if you're talking tubulars, the design means that the tire's dimensions (which can be different from what is quoted) reflect the diameter of a cross-section -- height or width. the width and height won't really change based on the width of the rim. for a clinker, things are a bit more variable.