Monoidal categories $\mathcal{C}$ which admit a monoidal natural isomorphism $\Phi_X:X^{**} \rightarrow \beta\otimes X\otimes \beta^*$ are called $\textit{quasi-pivotal}$ . For $\mathcal{C}=$Rep$(H)$ with $H$ a finite dimensional Hopf algebra, quasi-pivotal structures on $\mathcal{C}$ are in bijection with pairs $(l,b)$ where $l$, $b$ are group-like elements in $H$, $H^*$ respectively satisfying $$S^2(h)=b(h_3)\;b^{-1}(h_1)\; l\;h_2 \; l^{-1} \;\;\;\; \text{for all}\;\; h\in H.$$
Such a pair $(l,b)$ is called a $\textit{pair in involution}$. The paper Generalized Taft algebras and pairs in involution, constructs a family of Hopf algebras that do not admit a pair in involution, thereby providing an answer to the bonus question.
"Book Hopf algebras" provide an example of Hopf algebras that admit a quasi-pivotal structure but don't always admit a pivotal structure. See this paper for further details.
Furthermore, by a result of Shimizu, a quasi-pivotal structure on $\mathcal{C}$ yields a pivotal structure on $\mathcal{Z}(\mathcal{C})$. Thus, the Drinfeld doubles $D(H)$ of the Hopf algebras admitting a pair in involution (discussed above) provide examples of pivotal Hopf algebras.