I don't think that Cura is smart enough to cut STL files apart like that. My guess is that a program like blender would be best to do this, make a shape the size of your printer's build volume (or slightly smaller), put the part you want to print inside of the shape, then remove everything else.
I'm not familiar with blender but that's the work flow I would approach with.
Can even use cylinders to group some circular parts of the model together naturally so that the “glue” points are at the bottom inside the walls where they’re not obvious
Cura can ignore what's below buildplate as long as part isn't too big for X Y axes.
For cutting this, I would use 3d builder. Very easy to use. Just open model, and go to modify to cut model