Something is missing here. Not a part, but key info, or clarification.

Specifically, EXACTLY what is moving. OTOH you describe a pressed in cup. These are either a solid steel part, with bearing ground in, or a aluminum cup with a steel bearing race. If the latter, it would take more than an ovalized head tube to dislodge the race.

There are also integrated headsets where the bearing drops in, rather than needing to be pressed.

So, before going farther, we need some clarity.

What type of headset?
Is the cup moving in the headtube, or is something else happening?

The precision and clarity of your description will determine the quality of any help offered.