Putting aside what type of weapons are legal to sell & own, why is there such opposition to a national firearms database?
Fear of a policy change.
To take a different example, there are a lot of people in the US who are illegal immigrants, but were brought there when they were quite young. So what to do about them? Their presence in the US is definitely illegal, there is no question about that. But they were brought as children, sometimes as infants, so clearly the decision to enter the US illegally was someone else's. You sometimes hear them referred to as the "dreamers", although this term would usually only be used by those more sympathetic to their plight.
So the Obama administration began a programme that allowed them to register, which would make them ineligible for deportation for two years, and allowed them to work. Basically, they would still be illegal immigrants, but the government would simply ignore their illegal status. The registration could be renewed. There were some strings attached, if you have a serious criminal conviction, you're not eligible. The programme was called DACA, Deferred Action for Childhood Arrivals. The "Action" that is "deferred" is deportation. You're still illegal, but the government won't do anything about it for a while. This programme was not a law passed by a congress, it was an administrative action by the executive branch of the government, headed by the president.
Sounds great, doesn't it? If you're one of these people, instead of having to live an undercover, illegal existence, concerned that at any moment, your illegal status could be discovered, and the government could deport you, you could stand up and be proud, live openly, find a job, etc. You can't become a citizen, and you can't vote, but most of the ordinary everyday things citizens can do, you can do. Why would anyone not want to sign up for that?
The trick is, you had to register. If you weren't registered, and the government found you, you could be deported. And you couldn't just work a job whenever you wanted - you had to register first for the job to be legal. So you had to go to the government and identify yourself to be eligible. Then, there was a presidential election, and Donald Trump became president. Your name is now on a handy list of illegal immigrants in the possession of the government, whose head states openly that he wants to end this programme, and who in fact did attempt to end the programme. The courts ruled in June that the Trump administration has acted arbitrarily and capriciously in its effort to end this programme, but it did not rule on the merits of the programme, so we have the possibility that the Trump administration will try again to end it, in a way that will pass muster in the courts. (A similar thing happened with the so-called Muslim bans - the first two attempts were blocked by the courts, but Muslim Ban 3.0 was engineered to remedy the defects the courts found in versions 1.0 and 2.0.) Donald Trump's term is up in five months, but he might win reelection.
So you have the possibility that a president will no longer "defer" your deportation, and force you to go back underground. Except you can't go underground, because you went and identified yourself to the government as an illegal immigrant when you registered for this programme under the Obama administration.
So I think a lot of the gun people are concerned about something like that happening. Maybe there is no reason to fear being in a national registry right now, but if a future administration changes policies (and the courts allow it to happen), then registration means they have a nice handy list of everyone who has guns.