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.

  1. Step 1:

    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.

  2. Step 2:

    You need to edit the 3d sketch and cut the spline so it is not close the region you want to remain unaffected.

  3. Step 3:

    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.

  4. Step 4:

    In the deform region tab check fixed edges and insert as fixed the surfaces of the round shape you want to remain straight.

  5. Step 5:

    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.