Can I remove just the upper part of my 1 Piece Acrylic Tub (Walls) and leave the tub in place?
If you take a look at the picture in this question you'll see that a "normal" tub has an integrated flange that is meant to go behind the tile.



Your one-piece is not designed with tile in mind, so especially at the corners, you are going to have a gap between the tub and the wall.  Depending on the design of your one piece, cutting off the top might cause it to loose some rigidity and cause the tub to flex.

What you would need to do is cut off the top and leave a flange of sorts that you could use to attach the tub to the wall securely.  Then you're going to need some trim or possibly the "sill" that you referenced to make a good transition between the tub and the wall and tile down to that transitional piece.

It may be possible, but with the work involved and the unknowns - have you priced a basic tub?  Or looked at a building surplus type store?  I'd make sure you don't have any reasonably priced options before resorting to this (you can get a new, retail, enameled steel tub for less than $150).