How to bend my solid in the right way?
Here is a way to do it, although it will probably get more deformed that you like to.
First you will need to subdivide the part in order to keep the circular part straight. Draw a sketch like the pic and make a split line feature. Select all the faces of the part.
You need to edit the 3d sketch and cut the spline so it is not close the region you want to remain unaffected.
Create a deformed feature with curve to curve selected. Then select your initial curves on the first field and the 3d sketch in the second field.
In the deform region tab check fixed edges and insert as fixed the surfaces of the round shape you want to remain straight.
In shape options you will need to experiment to get the best result. Exit and see if your final shape is what you want. Good luck.