First I would break down the shape into eighths (1/8), so that I could mirror to get the final shape.
For each quadrant, I would draw conic curves with the ends being perpendicular to the primary planes with tensions to whatever looked good. One conic in the top view/plane, and another conic in the front view/plane.
Then I would sweep one conic, using the other conic as the guide.
I've never used Inventor, so I have no idea if Boundary is a tool the two programs share.
Boundary was added to SW a while back and is sort of like a better replacement for many operations that would normally be made with Loft or Sweep.
I don't think Sweep will work on this model unless some trickery is used to leave a tiny sliver of open space.
Loft is a method that works in SW too. I have not tried a Loft yet, but I did see another model using a Loft. It looks good. In SW there is a "close loft" option. It needs to be on.
I was actually surprised when the Boundary method closed itself up at the end. I thought for sure I was going to have 75% of the model. I tried a Solid Boundary as well and it works fine. I just usually default to working with a surface for anything that looks like this.
Not sure if it will help, but here's a screenshot of the Boundary tool in action. Each of the blue groups is a half ellipse type shape. The purple group defines the outside edge of the model. I have not tried adjusting the curvature controls yet (those flags that say "None").