It is true that in a cosmos $\mathcal{V}$ (= a complete and cocomplete symmetric monoidal closed category) with zero object where $2$ has a dual, the canonical map $1+1 \rightarrow 1\times 1$ is invertible. In other words, you don't even need to assume that $\mathcal{V}$ is locally presentable. I think that this implies that the cosmos in question is semiadditive, which in turn implies that coproducts/products are biproducts (the only thing one needs to observe is that $C\otimes 1\times 1 \cong C\times C$, which follows from the fact that $1\times 1$ is the dual of $1+1$).
Proving this is a bit too involved for MO (because there is no good way of drawing string diagrams here). I have typed up an argument, which can be found here (Wayback Machine). I first prove that an autonomous symmetric monoidal category (autonomous means that all objects have duals) where the coproduct $1+1$ exists is semiadditive.
I can try to give some intuition for the argument in question. The key idea is that one wants to construct a diagonal map $1 \rightarrow 1+1$. The way to do this was inspired by the following paper:
André Joyal, Ross Street and Dominic Verity (1996). Traced monoidal categories. Mathematical Proceedings of the Cambridge Philosophical Society, 119 , pp 447-468 doi:10.1017/S0305004100074338
In my writeup, this "diagonal map" is the map $1\rightarrow 1+1$ in the string diagrams which is built out of a "loop" and the map $1+1 \rightarrow (1+1) \otimes (1+1)$ which I called $h$. From the formula given in the introduction of the paper by Joyal, Street and Verity it follows that my construction does indeed give the diagonal map in the case where $\mathcal{V}$ is the cosmos of vector spaces over some field.
Edit: updated expired link.
