There is a bicategory of Dixmier-Douady bundles of algebras which is equivalent to the bicategory of bundle gerbes. In particular, sections into these bundles form algebras.
The price you pay is that the bundles are infinite-dimensional; for that reson I am not sure if that picture persists in a setting "with connections".
I do not know a good source for the bicategory of Dixmier-Douady bundles or for the equivalence. Everything depends certainly on the type of morphisms you consider between the bundles; they clearly have to be of some Morita flavor. You may look into Meinrenken's "Twisted K-homology and group-valued moment maps", Section 2.1.1 and 2.1.4. In Section 2.4 Meinrenken indicates indirectly that his bicategory of Dixmier-Douady bundles is equivalent to the one of bundle gerbes, by transfering the notion of a multiplicative bundle gerbe (which depends on the definitions of 1-morphisms and 2-morphisms) into his language.
Side remark: a bundle gerbe is not the direct generalization of transition functions of a bundle. There is one step in between, namely a bundle 0-gerbe: instead of open sets, it allows for a general surjective submersion as the support for its transition functions. If you take bundle 0-gerbes instead of transition functions, the functor you mentioned at the beginning of your question has as canonical inverse functor. See my paper with Thomas Nikolaus "Four equivalent versions of non-abelian gerbes".
Added (after thinking a bit more about the question): If you want to categorify the vector space of sections into a vector bundle, you first have to fix a categorification of a vector space. An algebra is one possible version of a "2-vector space", probably due to Lurie. Another version, due to Kapranov-Voevodsky, is to define a 2-vector space as a module category over the monoidal category of vector spaces (add some adjectives if you like).
Let us define a section of a bundle gerbe $\mathcal{G}$ over $M$ to be a 1-morphism $s: \mathcal{I} \to \mathcal{G}$, where $\mathcal{I}$ is the trivial bundle gerbe. Then, sections form a category, namely the Hom-category $Hom(\mathcal{I},\mathcal{G})$ of the bicategory of bundle gerbes (the one with the "more morphisms" defined in my paper which was mentioned in the question).
The category $Hom(\mathcal{I},\mathcal{G})$ of sections of $\mathcal{G}$ has naturally the structure of a module category over the monoidal category of vector bundles over $M$. Indeed, a vector bundle is the same as a 1-morphism between trivial gerbes, i.e. an object in $Hom(\mathcal{I},\mathcal{I})$. Under this identification, the module structure is given by composition:
$$
Hom(\mathcal{I},\mathcal{G}) \times Hom(\mathcal{I},\mathcal{I}) \to Hom(\mathcal{I},\mathcal{G}).
$$
The functor which regards a vector space as a trivial vector bundle induces the claimed module structure over vector spaces.
Summarizing, sections of bundle gerbes do not directly form algebras, but they form Kapranov-Voevodsky 2-vector spaces.
There is a canonical equivalence of $2$-categories
$$St\left(Man/M\right) \simeq St\left(Man\right)/M$$
between stacks on the large site of $M$ and stacks on the site of manifolds equipped with a map to $M$ (regarding $M$ as a representable sheaf). Given a map $\pi:\mathscr{Y} \to M$ for $\mathscr{Y}$ some stack on manifolds, it corresponds to the stack $\Gamma(\mathscr{Y})$ on $Man/M$ which assigns a map $f:N \to M$ the groupoid of sections $N \to \mathscr{Y}$ of $\pi$ over $f.$ Suppose that there is a cover of $U_i$ of $M$ such that each $U_i \times _M \mathscr{Y}\simeq U_i \times BU(1)$ (or if you prefer $U_i \times BGL(1)$). Then $\Gamma(\mathscr{Y})$ is easily seen to be a gerbe on the large site for $M$. By Dan Peterson's answer, we see that from the data of a bundle gerbe, one gets a stack $\pi:\mathscr{Y} \to M$ with this property. In fact, it is not hard to show that these are equivalent data, that is, given $\pi:\mathscr{Y} \to M$ such that there is a cover $U_i$ such that $U_i \times _M \mathscr{Y}\simeq U_i \times BU(1)$ is the same as giving a bundle gerbe on $M$. By taking each bundle gerbe $\pi:\mathscr{Y} \to M$ and sending it to $\Gamma(\mathscr{Y})$, one gets a fully faithful embedding of the $2$-category of bundles gerbes over $M$ into the $2$-category of gerbes over the large site of $M$ (which furthermore embeds fully faithfully into stacks on the large site of $M$). The essential image is precisely those gerbes on the site $Man/M$ which are banded by $U(1)$, as pointed out by Reimundo Heluani. It doesn't embed into the $2$-category of stacks on the small site of $M$ however.
Best Answer
The fact that you are dealing with compact and/or finite dimensional Lie groups is completely irrelevant. The fact that these group are Lie is also partially irrelevant (unless you care about putting connections on your bundle gerbes, in which case it becomes very relevant). More relevant is whether the groups abelian or not. A priori, the cocycle relation only makes sense for abelian groups.
But there is also a theory of non-abelian (bundle) gerbes, where you allow non-abelian groups. The cocycles have two kinds of data: Maps
$\alpha_{ij}:U_i\cap U_j\to \mathrm{Inn}(G)$ and maps
$g_{ijk}:U_i\cap U_j\cap U_k \to G$,
where $\mathrm{Inn}(G)$ denotes the group of inner automorphisms of $G$.
These non-abelian gerbes are classified by $H^2(-,Z(G))$, the second Cech cohomology group with coefficients in the sheaf of $Z(G)$-valued functions. [that's a non-trivial theorem]
That was the case of a trivial band.
A band is the same thing as an $\mathrm{Out}(G)$-principal bundle. Say you are given an $\mathrm{Out}(G)$ principal bundle $P$, described by transition functions $b_{ij}:U_i\cap U_j\to \mathrm{Out}(G)$. Then you can twist the above definition as follows: The cocycles now consist of maps
$\alpha_{ij}:U_i\cap U_j\to \mathrm{Aut}(G)$ and maps
$g_{ijk}:U_i\cap U_j\cap U_k \to G$,
where the $\alpha_{ij}$ are lifts of the $b_{ij}$.
The gerbes with band $P$ are classified by a set that is either ♦ empty, or ♦ isomorphic to $H^2(-,Z(G)\times_{\mathrm{Out}(G)} P)$, the second Cech cohomology group with coefficients in the sheaf of sections of $Z(G)\times_{\mathrm{Out}(G)} P$.
Whether or not that set is empty depends on the value of an obstruction class that lives in $H^3(-,Z(G)\times_{\mathrm{Out}(G)} P)$. It's non-empty iff that obstruction vanishes.
Finally, to answer your last question. If $G$ is a Lie group and you have a bundle gerbe with connection (trivialized over the base point), then you get a $G$-principal bundle, but only on a subspace of the based loop space $\Omega M$. It's the subspace consisting of those loops over which the band $P$ and its connection trivialize.