Is there a categorical version of the splitting principle? One of many places we see a "splitting principle" at work is in the category $\mathsf{Vect}(X)$ of complex vector bundles over a compact connected Hausdorff space $X$.   For any object $E$ in this category, we can find a bundle of compact Hausdorff spaces $p: F \to X$ such that
$$ p^* : \mathsf{Vect}(X) \to \mathsf{Vect}(F) $$
maps $E$ to a vector bundle $p^*(E)$ that splits as a direct sum of line bundles.
Given that this same idea shows up in many other contexts, I'm hoping there's a general version that subsumes a lot of examples.  Maybe something like the following.
Just for brevity, let's say a 2-rig is a symmetric monoidal $k$-linear category with absolute colimits.  (Such 2-rigs were studied in a paper I wrote with Joe Moeller and Todd Trimble, but there we were assuming the field $k$ had characteristic zero, and here I'd rather not—unless it turns out to be helpful.)  Having absolute colimits is the same as having biproducts and having splittings of all idempotents.  A map of 2-rigs is a symmetric monoidal $k$-linear functor; such functors automatically preserve absolute colimits.
I'm using 2-rigs to generalize categories of vector bundles: $\mathsf{Vect}(X)$ is an example of a 2-rig when $k = \mathbb{R}$ or $\mathbb{C}$, and the above map $p^\ast : \mathsf{Vect}(X) \to \mathsf{Vect}(F)$ is a map of 2-rigs.   (Note that $\mathsf{Vect}(X)$ is not an abelian category.)
Given an object $E$ in a 2-rig $\mathsf{R}$, I would like to find a map of 2-rigs
$$ f: \mathsf{R} \to \mathsf{R}' $$
such that $f^*$ is a direct sum of line objects, meaning objects $L$ with duals $L^*$ such that $L \otimes L^* \cong I$, $I$ being the unit object.
Are any general theorems along these lines already known?
Of course such theorems become less interesting if $f$ is "far from one-to-one".   It may be too much to demand that $f$ is fully faithful, though I'd like to if I could.  But we can define the Grothendieck ring $K(\mathsf{R})$ for any 2-rig $\mathsf{R}$, and in the vector bundle example
$$ K(p^\ast) : K(\mathsf{Vect}(X)) \to K(\mathsf{Vect}(F)) $$
is monic.   So, in the general case, a fallback position would be to ask for a map $f: \mathsf{R} \to \mathsf{R}'$ such that $K(f)$ is monic.  Are there general conditions under which we can achieve this?
I expect we'll need some finiteness condition to get $f(E)$ to split as a finite direct sum of line objects.  Furthermore, as pointed out by Simon Henry below, the splitting principle for $\mathrm{Vect}(X)$ as stated above fails when $X$ is not connected: this is a clue as to the further conditions we will need.  For example, it may help to assume that $\mathrm{End}(I)$ has no idempotents other than $0$ and $1$.   This is true for $\mathrm{Vect}(X)$ when $X$ is connected, but not otherwise.  For each connected component $C \subseteq X$, multiplication by the characteristic function of that component gives an idempotent in $\mathrm{End}(I)$, and splitting this idempotent we obtain a vector bundle $E$ that restricts to a trivial line bundle on $C$ and a 0-dimensional vector bundle on $X-C$.  Thus, when $X$ has more than one component, $E$ is not a direct sum of line bundles.
 A: I don't think there can be a reasonable such splitting principle, even in the weakest sense you asked about (i.e. $K_0(\mathsf R) \to K_0(\mathsf R')$ is injective) and in the case when the ground ring $k$ is a field of characteristic zero.
Let us make the standing assumption that $\mathrm{End}(\mathbf 1)$ has no nontrivial idempotent, as suggested in the question. Then if $L$ is a line object, we must have either $\wedge^n(L)=0$ for all $n>1$, or $\mathrm{Sym}^n(L)=0$ for all $n>1$. Indeed consider
$$ L^{\otimes 2} \cong \wedge^2(L) \oplus \mathrm{Sym}^2(L). $$
If both summands here are nontrivial then we get nontrivial orthogonal idempotents in $\mathrm{End}(L)\cong \mathrm{End}(\mathbf 1)$. The argument generalizes to higher $n$.
From this it follows that any finite sum of line objects is annihilated by some Schur functor. More precisely the sum of $n$ even and $m$ odd line objects is killed by the Schur functor given by an $(n +1) \times ( m +1 )$ Young diagram. This follows from Pieri's formula.
So let for example $\mathsf{R}$ be the 2-rig $\mathsf{Poly}$ of polynomial functors $\mathsf{Vect}\to \mathsf{Vect}$, and let $\mathrm{id} \in \mathsf R$ be the identity functor. If $f:\mathsf{R} \to \mathsf R'$ is such that $f(\mathrm{id})$ is a sum of line bundles then $K_0(\mathsf R) \to K_0(\mathsf R')$ can not be injective, since all Schur functors applied to $\mathrm{id}$ are nonzero in $K_0(\mathsf R) \cong \Lambda$.
