Buck, I would do the sistering, and I would screw into the adjacent joists and go all the way to the other wall side where that joists rests against the wall header.
This will assist you, but my concern is the span of those joists. Do you know what the span is? Is the tub to the side or middle of the above room?
I would think that you need to address the beam and post issue more than the sistering. For example: if the joists are spanning lets say 15 feet. that span was designed for normal above the joists weight, it was not designed originally for a 700 lb hot tub plus another 600 average pound occupants.
Sounds like to me you need to be thinking of a beam that is like a 8x10 that goes perpenducular to the joists and goes from one side of the room to the other and is supported on the sides by Kings and in the center or every 8 feet by heavy posts.
do you think that I am exaggerating the issue? Its better to think bad scenerios now rather than ponder them when the tub and people fall through the ceiling and landing on the first floor.