According to the FAQ, you shoud be able to switch over to the new ship. the only thing that is not retroactive if the Joined Tril. Once you claim it, I think all you will need to do is go to the shipyard and make it your active ship.
a few days ago, i would've said. you'll already be past the LT ranks so using a constitution ship would've been useless cause you would've surpassed the tier. BUT with all the down time since head start, i'm sure you can just redeem it.