The only DLC I've gotten for Dreamy Theater 2nd was Miku Append. I had already purchased it as a PSP DLC in July. When DT2 was released on Aug 6th, I noticed Miku Append was a free DLC for it as well. So I think you need to purchase the PSP DLC and then the DT2 DLC equivalent will become available for free. Someone please correct me if I'm wrong... Just make sure it's something that is available to DT2 before you purchase it for the PSP. Like Kasane Teto is not yet available for DT2, so don't purchase her PSP DLC expecting it to appear for DT2.
DLC in DT2nd is only avaiable, if you really purchased it in the PSN Store. Otherwise you have no access to it. So if you don't own your DLC from PSN Store, you can forget it... ^^"
As you can see there's a list of DT2nd DLCs in the in-game menu. If you purchase a PD2nd DLC which is DT2nd compatible, the DT2nd version will appear in the DT2nd Modules category in PS Store with free of charge. Idolmasters, Kasane Teto, and Extend module packs are not compatible in there.