If the cilinder is not a requirement, and you're open to a capsule, why not go all the way to an ellipsoid? That's how I did it 10 years ago, as back then I read that other games used that strategy too.
You transform the nearby geometry, by scaling it along the UP axis (renormalize the normals), using the center as the ellipsoid as the origin. Then your intersection math becomes SPHERE <=> TRI, which is trivial.
Once you found the intersection point, scale everything back, including the normals (renormalize again).