I'm pretty sure most FFLs will run the check on your buyer if you ask them to. I think the gov charges a small fee and the FFL might ask for a fee too but I'd bet most wouldn't. No need to put that on everyone to satisfy you.
I’m not saying background checks should be mandatory for private transactions. I’m pointing out that background checks should be focused on stopping nutjobs from getting guns, not attempting to create a national database of gun owners and the weapons they own (that’s what the left wants so it can confiscate weapons later on). If that system was readily available and easy to use for private transactions, then I’d certainly use it. However, there’s no way to enforce mandatory checks for private sales. Criminals will disregard them. If the goal is what the left claims, then I think there’s a reasonable compromise. Of course, the left is disingenious.