No - 'Commonwealth Realm', even if realm were in lower case, signifies a realm that is in the Commonwealth. That includes the realms like Swaziland and Tonga which are members of the Commonwealth and have a monarch other than Elizabeth II. Also, there is nothing in the term to indicate that Elizabeth II is the common monarch of these realms, and a term should descrtibe what it signifies. Belonging to the Commonwealth is quite incidental to the quality of the realms we are attempting to group.