Only songs from Green Day: Rock Band and Rock Band Blitz can be purchased as DLC, barring a few special exceptions. Painkiller is not one of those exceptions.
Theoretically, to export Rock Band 2's playlist, you would have to insert your copy of Rock Band 3, select Get More Songs, select Redeem Code, input an unused Rock Band 2 code, and pay the $10 export fee.
The problem is that the export codes that came with copies of Rock Band 2 have all expired, and while you can try to contact EA Support for a working code, they might not be able to send you a code, and even if they do, it might not work. The official comment is that the export is no longer available, although some have claimed success.