My thoughts ….....
Mary Elizabeth genuinely signed as widow Brooks on the death certificate, because that is how she saw herself. She would not willingly / openly admit to being a party in an adulterous affair. I don’t think she had any devious intentions when she informed on Joseph’s death.
A short while later, problems occurred with the military pension and final payout of funds – name and address of the real wife conflicted. Mary Elizabeth then was required to make amendments at the registry office, so everything was legally correct. We only have Joseph’s pension index card, as all other pension related papers, that might have told the full story, have been destroyed.
What is important here is that this is confirmation that M E Brooks and M E Harmer were one and the same person, and that she was Joseph’s partner. The signature on the 1921 census where we see Mabel changed to Mary endorses the fact that Mary was known as Mabel. (As I pointed out earlier, Mabel is a derivative of the name Maria. )
I think we have correctly identified the parents of Brad’s ancestor. The only way to be certain is through DNA. As Joseph Brooks and Mary Elizabeth Harmer both had children, there is the potential to do that.