Here a general abstract argument that shows that the subcategory of symmetric bimodules is never extension closed in case $A$ is a commutative finite dimensional Frobenius algebra that is not semisimple:
Let $A$ be such an algebra and $B=A \otimes_K A$ its enveloping algebra and assume that the subcategory of symmetric bimodules is closed under extensions. The simple module $S$ is symmetric and thus the subcategory of symmetric finite dimensional bimodules equals the module category of $B$. It thus also contains $B$, but $Hom_B(A,B) \cong D(A) \cong A$ has dimension less than $B$ and thus $B$ is never symmetric. This is a contradiction and thus the subcategory of symmetric bimodules is never closed under extensions.