Hi David - I've attached here a version with a cleaned up ellipsoid that should now work properly.

I did an "untrim" (selected the boundaries and deleted them) of the doors to get them back to their full surfaces, and for the ellipsoid I extracted the edge of the ellipsoid and did a new revolve, making sure to snap the ends of the revolve axis exactly on to the ends of the generator curve.

Now you can use the Trim command to cut the ellipsoid and the door at the same time.

To do that, select both the door and the ellipsoid. Then run the Trim command.

The command prompt in the upper-right area of the window will say "Select cutting objects or Done for mutual trim" - in this case you want to make each object both cut the other and also be cut itself, which is the "mutual trim" method. So at this stage of the Trim command just push Done or right-click in a viewport to tell Trim that you are not picking any other cutting objects.

Then there will be a pause for a moment as it calculates the fragment pieces, and then the next stage of Trim you can either just push Done to keep all the pieces, or you can pick the pieces you want to have discarded. Sometimes it is easiest to just have it slice up all the pieces and then delete the ones you don't need after Trim is finished. If you do it that way then you can use Join after you have finished deleting things to glue things together.

- Michael

EDIT: Same thing Burr is saying above...