Is there another way to "weld" physics enabled static meshes together, or am I using constraints wrong?

Sort of. You can weld objects together by making them a child of a singular parent object and then setting “Auto-Weld” to true in the physics category. There will be no wiggle at all.

However, all child objects will inherit the weight and physics material of the parent object. And as far as I know there is no fix for that. So it’s not perfect.