Want to install Japanese style unit bath/bathtub. Trimming a joist a bad idea?
Short answer is no way.   You aren't trimming you are taking out the joists there.  It wouldn't even work with a 20 pound tub and a 100 pound person.    Also if it is a no the step option shouldn't be an option.   I have seen this on many a house and what the hell do you do at the top of the step, dive in the tub?   I am rather tall and have dealt with it, can't imagine someone shorter or older.   Not to mention making this look luxurious is almost impossible with the step method.   I have never seen it done where I was like &quot;wow that looks nice&quot;.
Long answer yes you can do anything with enough money.   If your joists are parallel to the tub it is a bit cheaper (probably) and you can simply have an adequate header installed to carry the joist load.   Now I have no idea what is below this part of your basement but you may need a pole or a wall or whatever.   With enough money thrown at it an architect can design a way to carry the load for sure.   This could range in as little as 2k up to 10-12k (plus maybe an out of place wall/pole).
If I had already bought this and can't return it, it would unfortunately go in my basement.   I can't see paying for something like that, having it installed quirky, and then effecting other parts of my house.