I feel like outdoor-only basement access makes sense for a basement you're going to use as storage, not a basement that's part of the living space.
I'd probably put a 1-tile-wide staircase down to the basement either in the hallway (using 1 of the 2 tiles of hallway width) or running down the length of the current laundry room and bathroom, making the new bedroom a little more narrow. And then move the basement however you like to suit the staircase positioning.
Purple boxes show those two possible staircase locations
https://i.postimg.cc/QdPQxchC/06ZXRia.png