Before doing anything, you may want to have a structural engineer check this. My place was originally designed so the chimney helped to support the main beams. (That eventually failed, either as part of or causing the subsidence of part of the house; that weight is now on a Laly column.) You don't want to take out something holding your house up.
And you don't want to in addition place the weight of the chimney on members which may not be able to carry it, which is what taking out just the bottom would do. If you really insist on that approach, a structural engineer could tell you what needs to be reinforced to survive that new load. No guarantee it's possible.
The house is probably the most expensive thing you own, other than the other Hunan residents. Don't gamble with it.
The counter-suggestion to demolish the whole chimney, working from the top down, makes far more sense to me. Since my own chimney is no longer in use (direct-vent furnace with both intake and exhaust on the side of the house), I had the top of my chimney taken off when I redid the roof, and then did my best to seal the top and bottom so I wouldn't lose basement heat into the attic through it. I would like to take it out completely someday, but having removed a few bricks as an experiment, I really would prefer to leave that job to pros. And while recovering those cubic feet would be nice, I don't really need them, or the hassle involved.