Don't filibuster the back ground check. The left will just use it to beat you over the head. Instead attach things like Exemptions for guns given or sold to family members or friends of more than 2 years that you know are not felons. Make the Gov. pay for all BGCs with a shall not charge clause that states that upon demand any police/sheriff dept. must at not charge to the buyer or seller do a BGC for a gun transaction.
Couple this with a clause that states that if any police / sheriff dept denies such BGC that the buyer and seller are free to do the sale with no repercussions to the seller of any legal sale.
Also maybe some sun down clause that says that a study after 10 years must be done at the expense of the federal government and if it does not show more than what ever percent of crime stopped then the laws sunsets.
This way its not killed outright but it does not burden the gun owner who wants to give a gun to their wife husband, son or daughter, ect.