Does the funeral home have a right to arbitrarily decide that someone other than the eldest living child will instead be the next head of the family and then interject into the Rite of Committal an oath made under God binding the family to this person of their choosing?
No, absolutely not – the funeral home has no authority to make such decisions for your family. This is entirely between your family and your priest. It sounds like this was a serious misunderstanding or mistake by the funeral home. I am very sorry this happened and I would recommend that you speak to the funeral home manager about it, along with your priest. If only, at this point, to ensure they understand the error.