No.
They would have to prove that I bought it after the BGC law went into effect.
If it was a firearm that was bought by the seller before the BGC law went into effect, then they would have to find the seller and have the seller testify that they sold it after the law went into effect - which a seller who skirts the law would probably be reluctant to do since they would most likely be in more trouble than the buyer.
Could they find the seller? Maybe, maybe not. If the seller had bought it from an FFL then sure, but if the seller had bought it from another private party, who had bought it from a private party, and so on, it would be difficult. I don't keep track of who I sell my firearms to most of the time - not over time - and I am sure that at least half the firearms I have sold have probably been in turn sold to someone else.
Which gets to the crux of the issue; such laws, when they involve the serial # of the firearm, are more about firearm registration than about BGCs.
And what purpose would a government entity have for registration?
Confiscation is the only reason I can think of.
Registration doesn't stop gun crime. You can't reliably trace a firearm used in a crime by the projectile - it is just too easy to change barrels on many guns without any record of purchasing the parts to do so since those parts are not considered a firearm themselves - even to the point of changing the caliber, and you would have to have a suspect to start with and/or have a projectile from the same firearm used in a different crime previously.
You can't by the firing pin marks or the shell casings for the same reasons.
Eventually you have to have a suspect and recover their firearm and hope that they were stupid enough to not use a different barrel/etc. during the crime. Plus, a person can just say their firearm was lost or stolen (only 11 states require such reports, OR & WA are not among those states).
Plus, government entities simply do not care that much about solving crimes by these means. What they do care about is the balance of power between the citizens and the government itself. Eventually it will come around to firearm bans and then confiscation. Connecticut is already there and one could argue New York and California among others, are also.
Gun owners in Conn basically told their legislature to shove where the sun don't shine.