  We prove that each exponential functor on the category of finite-dimensional
complex inner product spaces and isomorphisms gives rise to an equivariant
higher (ie. non-classical) twist of $K$-theory over $G=SU(n)$. This twist is
represented by a Fell bundle $\mathcal{E} \to \mathcal{G}$, which reduces to
the basic gerbe for the top exterior power functor. The groupoid $\mathcal{G}$
comes equipped with a $G$-action and an augmentation map $\mathcal{G} \to G$,
that is an equivariant equivalence. The $C^*$-algebra $C^*(\mathcal{E})$
associated to $\mathcal{E}$ is stably isomorphic to the section algebra of a
locally trivial bundle with stabilised strongly self-absorbing fibres. Using a
version of the Mayer-Vietoris spectral sequence we compute the equivariant
higher twisted $K$-groups $K^G_*(C^*(\mathcal{E}))$ for arbitrary exponential
functor twists over $SU(2)$, and also over $SU(3)$ after rationalisation.
