Not an answer, just an illustration to accompany the question.
$K$ is an isosceles triangle with base $2$ and altitude $3$.
The ellipse $E$ with the (approximately) smallest area symmetric difference
has center near $(0,1)$ and radii about $0.667, 1.155$.
All ellipse parameters approximate.
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[![SymDiff13][1]][1]
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  [1]: https://i.sstatic.net/Sk1mH.jpg