Every right listed in the Bill of Rights has it's limitations, including the 2nd Amendment. Which is why it is illegal for you to own bazookas, machine guns, etc. There is a federal law that requires background checks prior to purchasing a gun from any retailer, and so far that law has not been struck down. Meaning that while that is not specifically stated in the US Constitution it is nevertheless in effect. You might not like it, but that is a fact.
OK. Let me present my US Constitution short course once again. Yes, it's a long post. Take the time to read every word of it and you'll come out of it at the end being a smarter person.
The government of the United States was created by the Constitution. It is a new government, new nation, from that date; previous governments formed under the Articles of Federation being dissolved by the enactment of the Constitution. The power of the government is both created and limited by what's in the Constitution. The government has no power not granted it in the Constitution.
Many people, and I'll tie this to the current discussion in a bit, misunderstand the government and the Constitution to claim things such as that only citizens get the rights or protections in the Constitution or that foreigners in the United States are not allowed to keep and bear arms. Or they believe that prisoners in Guantanamo can be treated differently than the Constitution allows because it's outside the boundaries of our nation, or they believe that the government does not have to follow the Constitution in dealing with foreigners on foreign lands.
In all these cases, they would be wrong with one exception, rarely implemented, and not once called upon since June 5, 1942. That is the last time Congress approved a declaration of war. In a war, the laws of war rule when dealing with the enemy of that war and the Constitution allows the President to engage in a war. Of course every other foreign engagement since then has been wholly unconstitutional.
So let's talk about why all this matters and why it matters in the case of gun background checks and registration.
The Constitution gives the Federal Government certain powers and it limits certain powers of the Federal and State governments. Out side of those enumerated Federal powers, there actually IS NO governmental authority. This limit is 100% absolute. The Supreme Court can't grant the Federal Government power that the Constitution did not give the Government or the Court. How can they simply make up power they don't have?
So when the Government, or people who happen to be on the payroll of the Government for their day jobs, wants to treat anybody, citizen or not, on US soil or in or about the embassy in any foreign country, or anywhere else, in any way not authorized by the Constitution that created the Government then they are no longer the Government. Under what authority would the Government deal with any person in any way not authorized by the Constitution, no matter where in the world the person was? When they step outside of their governmental authority they are no longer the government.
So, understanding that those people are nothing except people when they operate outside of the Constitution. The Constitution does not say that the Government can do anything outside the Constitution when dealing with non-citizens or that the Constitution only applies to citizens. It doesn't say that the Government is not limited to the Constitution outside of the US territory. How can it say such things? The government only exists within the limits of the Constitution, ever, in dealing with any person, in any place. Please explain if you think they have other powers than that and how they got those powers.
All of the above is about making one thing clear: The Government's power is exactly what's granted them in the Constitution and there is absolutely no other powers that they can claim for themselves under any circumstances.
So let's get down to your view, and Biden's view, that there are limits to the limitations on the government in the Constitution. You both claim the limits are on the rights but the Constitution didn't grant any rights, it limits government's ability to violate those rights. There's no right to free speech defined in the Constitution; the Constitution simply forbids the Federal Government from abridging the right to free speech. Many will claim the crowded theater exception proves there are actual limits but that;'s a lie that most Americans have swallowed as the truth. First off, no one can reasonably make the case that free speech as intended in original intent, included such a thing; it was clearly political speech, religious speech, etc. that was intended. And yelling fire in a theater should never be against the law either. It would be, at most, a civil tort, where the person would be sued for any damages or injuries; there's no need to make it illegal.
Similarly, the 2nd Amendment says "shall not be infringed." There are no exception clauses and no one can provide any constitutional reference to suggest that there are exceptions. There can't be any exceptions because, as I discussed earlier, outside of the Constitution, the Government has absolutely zero authority. If it's not written in the Constitution, then the Government just plain can not do it.
To argue that it is different, for instance limits to all rights as claimed by you and Biden, then you're suggesting that there are times when a person can be imprisoned for a crime without a trial (oh, yeah - think Jan 6; you most likely do support that). Others here have defended Terry stops in violation of the 4th Amendment. But those exceptions don't exist either. It just plain is NOT possible for the Government to operate as a legitimate government of the United States outside of the explicit, enumerated, powers in the Constitution.
So many make reasonable (though incorrect in my opinion) that it's a good thing to prevent felony litterers from owning a gun, or a Naval officer kicked out of the Navy for having an affair - a limitation not applied outside of the military. OK, we disagree. But to make those laws constitutional requires a constitutional amendment.
There is absolutely zero authority for the Government to engage with any human being, in any subject or thing, in any place in the world, outside of the powers granted in the Constitution. Period. No exceptions to the rights because the rights are not created by the Constitution. No exceptions to the limitations because the Constitution allows for no authority beyond the enumerated authorities. Neither Congress, the President, nor the Courts get to simply make up power not in the Constitution.