After importing the .STL into Blender...
Create a lattice scaled and positioned to contain the object to be deformed, and adjust the lattice UVW values to give it enough control density. (UVW is under object properties specific to a lattice.) Then give the object to be deformed a lattice modifier. Now go back to the lattice, and give it a simple-deform bend modifier. You may also want to use an empty as the reference point for the bend, as that can provide more control.
The intermediate lattice gives more control at preserving volume than the simple deform modifier alone. You can move lattice cage points in edit mode, and it'll affect deformation prior to the simple deform modifier which is applied to the lattice. That in turn should help with controlling how the deformation is applied.
Thus you should be able to adjust your object with those things in reference to a cylinder mesh.
Now you should be able to export that back out, and toggle the check in export settings for modifiers to be applied before export.
If for some reason the lattice doesn't deform strongly enough or has some issue of the object falling out of the lattice bounds, then it may also be worth looking into the mesh-deform modifier as an alternate. It works in a similar way to the lattice modifier but with a bind system, so shifting the control mesh after binding doesn't present some issues of boundaries that the lattice will.