I don’t know the specifics of the ban, but Oakmont has every right to ban him from the property. They have no right to ban him from playing the US Open. If he’s qualified and eligible to play per the USGA and Oakmont insists he not play in the US Open, the USGA should tell them to get bent and take the tourney elsewhere.
This is a case of Oakmont getting awarded those future US Opens, getting big in the head and not knowing how to act with their newly acquired perceived power. It is petty, classless, and disrespectful to the British Open to publicly (through their members) announce this on the eve of the Open.
Augusta already provided the playbook to ban an unwanted participant… you simply send him a letter. The fact Oakmont went out of their way to get in the limelight while Augusta goes out of their way to stay out of the limelight tells you everything you need to know about anyone who wants to come Augusta to Oakmont.