This group $G$ (or its close relative GL$_2$ over the same field) has been long studied and is pretty well understood.   But you have not invoked the characteristic $p>0$ in your question, whose answer depends a lot on how $p$ is related to $r$: sometimes $\nabla(r)$ is simple but sometimes it's a more complicated indecomposable module.  In any case the composition factors of your tensor product can be sorted out using recursion and the linkage principle.  

Recall that in characteristic 0, these symmetric powers $S^r(E)$ have for almost a century been used as models of the irreducible representations of the corresponding rank 1 algebraic (or Lie) group.   Such a space just consists of the homogeneous polynomials of degree $r$ in two variables, and has dimension $r+1$.  Lie theory assigns to $S^r(E)$ the highest weight $r+1$ times the (unique) fundamental dominant weight $\varpi$, and tensor products are complete reducible (Weyl) with multiplicities given by a Clebsch-Gordan rule.

In the modular situation the dual Weyl module you denote $\nabla(r)$ is simple (affording an irreducible representation) for $0 \leq r < p$ but then indecomposable with an increasingly complicated (but well understood) structure.  Here the simple module with highest weight $r \varpi$ is the unique simple submodule but due to Steinberg's tensor product theorem its dimension is usually $< r+1$.  When $r \geq p$ other linked weights may also occur.    

As an extreme example, the (first) Steinberg module $\nabla(p-1)$ is simple of dimension $p$ but has an *indecomposable* tensor product with $E$.  This has three composition factors.  In such a case your short exact sequence can't split.  Indeed, this indecomposable $G$-module of dimension $2p$ is a lift of a projective/injective module for the Lie algebra (or first Frobenius kernel).   For details in rank 1, including the way these modules restrict to finite Chevalley groups, look at my old 1973 J. Algebra paper (link from my homepage) or the more comprehensive treatment in Jantzen's book.