You will most certainly need to move the tub drain. That move may very well be more than just a inch or two. There are three issues I see here.
- Non-uniform dimensions. As in the diagram below the green dimension on a bath tub is typically longer than the red dimension. Then depending on the top side shelf widths the green dimension may very well likely not be centered on the tub overall width. This means that if you can rotate the tub ninety degrees the drain will not be possible to be the center of rotation.
The second issue is that built in tubs have one side face that comes down to the floor. The tubs can be purchased with the drain hole on the left or right side when the side face of the tub is in front of you. If you intend to reuse the same tub you will end up having the drain on the opposite end that what you envision with a simple rotate. If you intend to replace with a new tub you will have to make sure to switch the leftness/rightness of the drain hole location with respect to the current tub.
From your pictures where you depict the toilet located on the second floor it seems very likely that there is plumbing coming down from the upper floor to the lower floor in that stub wall that is in the lower level bathroom. You will want to check this carefully before you finalize your plans so you can factor in what this may mean to your overall project.