Try to buy the DLC cars from the dealership on your second account. You should be able to if you have enough in-game credits, but in any case you shouldn't get a message telling you to buy the DLC from the PSN store. My guess is that only the account which purchased the DLC cars with real-world money get the cars for "free" in-game, while other accounts on that PS3 need to buy the cars with in-game money in order to get them.
As for the tracks, did you buy the track packs? If you did, I don't know why Spa, Kart Space and Route X aren't showing up on your second account.
The Family Pack is only necessary for the DLC which came out back in October (Spa, Kart Space, the X2011, the three karts and 11 touring cars)