I don't see how dragging things through a lengthy legal proceeding is going to ease their pain. Rather, I would think that it would prolong it.
A graveside service, installing the headstone with 'Gwen' on it would go a lot further, IMHO.
There's no lengthy legal proceeding in the offing here. The family submitted a paper filing to the court, possibly accompanied by an extremely brief courtroom appearance. The judge will have his clerks do a little research to see if there's any applicable statutory or case law that needs to be considered, and then he'll issue a paper ruling, possibly accompanied by an extremely brief gavel-banging in the courtroom. Lengthy proceedings require 2 opposing sides, and unless some vicious religious nuts file an appeal to block the judge's (probable) decision to allow the change, there won't be anything to proceed about.