Is posible shaping concave bottom with profiler machine ??
Why? Once you have the bottom shaped by the profiler adding concave is no big deal.
I think it is the same as how you would make a crowned deck, but in reverse. You would need to make adjustments to get the depth right, but I think the guide that you use to make the cut across the bottom would need to have the concave curve.
Seems like a lot of work just making the guides. If you plan to make a lot of the same design, it may be worth it.
I dabbled with a shaping jig that used another board for the dimentions. Copying jig would be more like it. Took it apart after I used it once.