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This is related to the splitting of the unitary group $SU(B, \sigma)$, a topic which is addressed in "Generic Splitting of Reductive Groups," by Kersten and Rehmann. Specifically, Corollary 6.3 produces a field satisfying which seems to satisfy your desired properties.

I would think that you could take the function field of $R_{E/k}(SB(B))$, the Weil restriction of the Severi-Brauer variety of $B$. That would definitely split $B$, but I am not sure why you would still have a field when you take the tensor product with $E$. (This may be what they do in the paper I cited, but I couldn't follow the notation.)
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This is related to the splitting of the unitary group $SU(B, \sigma)$, a topic which is addressed in "Generic Splitting of Reductive Groups," by Kersten and Rehmann. Specifically, Corollary 6.3 produces a field satisfying your desired properties.

I would think that you could take the function field of $R_{E/k}(SB(B))$, the Weil restriction of the Severi-Brauer variety of $B$. That would definitely split $B$, but I am not sure why you would still have a field when you take the tensor product with $E$. (This may be what they do in the paper I cited, but I couldn't follow the notation.)