I'm pretty sure I know the answer to this already but I thought I would ask.

If I play this strategy card and I get to choose between Imperium Rex and another objective and I decide to play imperium rex, can I still claim an public objective that the Bureaucracy card says I can?

No, you can't. Imperium Rex is an immediate game over. Since you are told to claim an objective after revealing one, you lose that chance on revealing Imperium Rex. If you are tied then you break out the Tie Breakers (first number of planets then CCs in reinforcements).

digdoug is 100% correct. Imperium Rex, when revealed, always ends the game immediately. This is true via Bureaucracy as well as the original Imperial card.