Yes there is an easier way. An ellipse is only 4 circular segments which could easily be cut with g02 g03 once you figure out the geometry. I learned to do it in drafting class in school. Take a look at the attachment. You can see how the ellipse is constructed from the four circular segments. The centers of the two small circles are found by drawing a line from the top and bottom vertex of the diamond to the midpoint of the lower segment. The only tricky part may be finding the size of the leg of the diamond (all 4 are the same length) and the angle it is at. I think all ellipses can be constructed this way. You may be able to use some tangent relations to figure out the size of the diamond. After you have done that, it is simple to get your start and end points and circle center to program in a g02/03.
Edit: this is called stevens method, as seen here: