>ultimately any kind of constitutional convention would strengthen gun rights
You actually believe that an event where the ENTIRE constitution is officially and lawfully open for editing in its entirety will be beneficial? It would be worse then an election year where every state's legislature is effectively a battleground to determine the selected representative. Use the following source https://www.270towin.com/content/state-government-trifectas
to see which states are completely controlled by one party or the other. If a convention was called today and assuming 1 representative per state, there would be 22 Republican and 17 Democrat representatives from the start. The remaining 11 would not guarantee the 38 states/representatives required for a 3/4th majority for the Republicans nor the Democrats.
Given the track-record where the Republicans are too afraid to actually have a battle over the debt ceiling, and a requirement for compromise simply to end the convention without any changes, then it becomes abundantly clear this is an extremely shitty idea. Any changes or additions will massive repercussions and are almost guaranteed with a need to compromise. But of course I am a pessimist in the extreme, especially when it comes to touching the constitution. I certainly wouldn't want to see a section of the commitment to supporting Israel or the guaranteed support of "diversity" or addition of new pronouns because every state, no matter how retarded, is now in the game when a convention is called.
Article V of the constitution, which is partially about creating a constitutional convention only states:
>...on the Application of the Legislatures of two thirds of the several states, shall call a Convention for proposing Amendments, which, in either Case, shall be valid to all Intents and Purposes, as Part of this Constitution, when ratified...by Conventions in three fourths thereof
In regards to the conventions themselves. Since the creation of the constitution one has never been called, one can argue endlessly if limited conventions are allowed or not. In practice, the constitution was created from a convention that was initially about the articles of confederation, so there already exists precedent for changing the scope of the convention. If it occurred today, it would most likely be up to the representatives of the convention to effectively block proposals and force the scope to be limited.