You do need to be high enough level to claim the ship in order to get the Console - assuming you're getting it from the T3 Akira. If you can't fly a T3 ship you can't claim it.
STO is about my Liberated Borg Federation Captain with his Breen 1st Officer, Jem'Hadar Tactical Officer, Liberated Borg Engineering Officer, Android Ops Officer, Photonic Science Officer, Gorn Science Officer, and Reman Medical Officer jumping into their Jem'Hadar Carrier and flying off to do missions for the new Romulan Empire. But for some players allowing a T5 Connie to be used breaks the canon in the game.
If you've bought the Legacy Pack? You should be able to claim it once the trial are over or you've skipped it.
If you only have bought single items, you need to wait for the respective level restrictions to be fulfilled. The ships cannot be unpacked until the required level are reached. All you might be aware of, but just making sure.
If you however have trouble in the range of complete store failure? I would recommend a ticket.