Seems like you can simply use “Flow along curve” and use a straight line as a base curve, then use your existing elliptical curve as the target. That will “bend” the entire part.
However, if your goal is to extend only the end surface so that it takes the shape of the curve, then you can do that with the “Move face to a boundary” command. First, you have to extrude the curve in the same direction as the end surface of the solid model (the most convenient way to do that is to temporarily set the CPlane along the surface you want to modify with “’_CPlane _Object”). Then, run “Move face to a boundary” and select the face you want to modify (I coloured it in yellow), then select the extruded surface (coloured in blue).
An alternative way is to move that face further, by holding simultaneously Crt+Shift (or use “Move face” with “DirectionConstraint=Normal” from the command line) and drag the end surface with the Gumball until it moves enough to overtake the extruded surface that I mentioned above. Then run “Boolean difference” and use the extruded surface as an input. For this particular case you don’t need to do anything else, but if the extruded surface had opposite direction you can swap it either with “! _Dir”, “! _Flip” ор “! _ShowDir”.