I couldn't open your CATPart file although I could see it in miniature as a PreView, but I saw enough to work with. Your V5 Release is probably higher than mine (R9) which prevented it opening.
So, I modelled a quarter of the closed Guide Curve as a Conic and put in Section Curves (Splines) at each end of it. These met above the centre of the model. Extruded surfaces were made from the splines in a normal direction from the planes they lay on. These extruded surfaces will be for Tangency.
Then using Loft the two splines as sections, the extrusions for tangency and the Conic as a Guide a quarter of the Final surface was made. Mirroring this and Joining up afterwards gave the Irregular Dome which has been emailed to you. Using FreeStyle Shape Analysis, it doesn't look too bad. Ive only used the end sections but more can be incorporated if required.
There are other methods but try this for a start, it's simple.