Is wearing a patched garment and sleeping in it while it is knotted, then giving it to a pilgrim (mu'tamir) to untie the knot with the intention of overcoming spinsterhood and accelerating marriage, considered forbidden (haram) or does it fall under the category of shirk (polytheism)?

This act is impermissible and is among the ways of charlatans, because whatever benefit is not known through the Sharia, nor through experience and custom, it is not permissible to believe in its benefit. Charlatans are not to be trusted in this regard, nor are their instructions to be followed. The people of knowledge have stated that a person's reliance on something as a cause, unless Allah has permitted it legally (shar’an) or perceptually (ḥissan), is a minor shirk (polytheism), as Sheikh Ibn Uthaymeen clarified. And causes that have no effect according to the Sharia, custom, or experience are of no benefit to a person.