The animation produces a constant point amount change, I wouldn’t call it simple. Many export formats can deal with such.
Cappucino bakes PSR as well as PLA. PLA means here points of Polygonal objects only.
Splines are tricky, as the values are relatively complex to export. This is twofold. When you set up a Bezier Spline, check out the Structure manager. While that might be easily storable, the resulting intermedia points might not. Since each application uses its own idea of interpolation, the results might differ. FBX has obviously no ability to store this as animation.
FBX was developed as a format to store motion capture data, which means its main focus was animated joints. It is widely used, but that doesn’t mean it is the best to use. I assume the one asking for FBX has no other options for import? The question would be what is the target app, and what other option has it.
The top Extrude/Spline combination doesn’t work because the Close Spline option is off.
Both shapes work just fine with Alembic as an export format.
The OBJ format functions as a sequence but leaves the PLA interpolation off as the point amount changes.
I have changed the caps to Delaunay, as N-Gons are typically not a good idea to export.
I hope that either Alembic or OBJ sequence might help.