Thanks for the included pictures; they ensure the visually imagined situation is in keeping with what you are dealing with.
The existing tub should be able to come out by rolling it sideways (onto it's finished side) out of it's niche once you've detached the trap & removed any further obstructions such as the faucet.
To avoid removal of the faucet, you can also free the tub by removing the trap, then raising the finished side of the tub with prybars, slipping in a couple 1X boards underneath as skids, then sliding the tub out of it's home, where it can then be turned on it's side and transported out the doorway. The space is designed to fit closely to the tub, so you'll have to watch out to not get one end too far ahead of the other.
Be sure you've grabbed "a few good men" (or women; no politics here) to help with the removal process; those cast iron tubs can weigh quite a bit.