@ dollars
Regarding Queen - why is that necessary? Can you please please tell me where the constitution, other than Section 2, references the Queen as anything other than "the Queen"? And why, if the UK can restyle HM's titles we are not permitted to do so as well? And finally, how her style or title is relevant.
Regarding Letters Patent - much seems to be made of this by various writers that somehow the revocation, or non-revocation, or non-reissuance, or reissuance in the wrong manner in the opinion of the particular writer has affected the legitimacy of certain appointments. Which one it is seems to vary depending on the point trying to be made. So as far as I have been able to glean across various sites, the issue is that even though there are/were letters patent, some authors do not accept their validity. So to address your question, if they were produced, it would not change that author's mindset, that the letters patent produced were not valid. How does one address that when the other party refuses to accept that the document signed by the then monarch is valid? I can argue (and apparently on this site still am) till I'm blue in the face but if another refuses to concede that we are at stalemate - for example, whether my Commission as an officer in the Army is valid since it was signed by a GG who was in their opinion, not validly appointed by HM by letters Patent in a form which they believe is the "right" form, regardless of the fact that there is no "right" form.
I guess what I am saying is, even if Letters Patent were produced, some people will then argue "oh but those are not valid since it was signed in this spot rather than at that spot" or "it's not valid since the seal wasn't affixed this way, it was attached that way" (which arguements I have seen btw on sites). So producing them will achieve nothing. Could they be produced, I'm sure they could, would they be accepted by the party demanding their production, I'm pretty sure they would not.
But I am pretty sure if you asked Elizabeth II if she signed Letters Patent on 21 August 1984 she'd say yes that's my signature, even though according to some it's in the wrong place on the document and therefore invalid.
As for him/her, there has been a long standing acceptance in the interpretation of laws, sometimes codified eg Acts Interpretation Act, that references to male include female, single includes plural etc etc, should the context so require. Acts have also always been interpreted such that a reference to "his Majesty" includes or also means "her Majesty" etc