No, that’s not the reason. Four bishops, by your count, have called for more stringent gun laws. None of the others have. Stop saying it’s “the bishops” as if to include them all. It’s a maximum of four out of what, 534?
How many people are losing their lives because of private sales at gun shows? You have never told us that statistic. Until you do, the argument is empty.
Requiring that I do a background check if I sell my rifle to my cousin is simply designed to prevent me from doing it at all, because I do not have access to criminal records. So what you’re saying is that for me to do it I have to pay a licensed gun dealer to run a background check on my cousin or friend or my son if i want to give him my gun. And for that, the dealer will charge what?
In the absence of showing a compelling reason why I or any other member of the public ought to be obliged to go through that, the argument in its favor simply fails. Nobody here has shown that there is any.