I have a repeated problem with beveling complex meshes using the respective modifier, even if all what I do is to combine off-the-shelf geometric primitives with off-the-shelf modifiers, without any individual vertex-level manipulations. The effect looks often like that:
I tried to change the bevel modifier's many options, but the effect is that either the bevel is gone completely or the glitches become even larger.
To work around this problem, I finally replaced the second bevel with a lattice and it even gave some more control on the shape.