In order to take their seat in Parliament and draw the MP salary, all MPs must swear:That's not true. The monarchy exists with the consent of Parliament and it could abolish the monarchy with a single Act. Even Republic know this. https://www.republic.org.uk/abolish_the_monarchy
I swear by Almighty God that I will be faithful and bear true allegiance to His Majesty King Charles, his heirs and successors, according to law. So help me God.
Also, take a look at the Parliamentary procedure Erskine May:
Erskine May Paragraph 22.15 said:No question can be put which brings the name of the Sovereign or the influence of the Crown directly before Parliament,1 or which casts reflections upon the Sovereign2 or the royal family. A question has been altered at the Speaker's direction on the ground that the name of the Sovereign should not be introduced to affect the views of the House.3 Questions are, however, allowed on such matters as the costs to public funds of royal events and royal palaces