"Tensor complement" of representations of finite groups

According to my rather amateurish (so it would be good if somebody checked independently) GAP calculations, this is false for $G=SL(2,5)$ and $V$ either of the $4$-dimensional irreducible representations.

In fact, more is false. There are no representations $W_1$ and $W_2$ so that $$R\oplus (V\otimes W_1)\cong V\otimes W_2,$$ so if $V$ has character $\psi$ and the irreducible characters are $\chi_1,\dots,\chi_n$, then the regular character is not an integer linear combination of $\psi\otimes\chi_1,\dots,\psi\otimes\chi_n$.

The $SL(2,5)$ example is probably small enough to verify by hand, but it can be done in GAP using only built-in functions, with no non-trivial programming. GAP has functions to do things like constructing tensor products of characters and calculating scalar products of characters, so it is easy to construct the matrix $A=(a_{ij})$ such that $$\psi\otimes\chi_i=a_{i1}\chi_1+\dots+a_{in}\chi_n$$ for each irreducible character $\chi_i$. Then it remains to check, for $b=(b_i)$ the row vector such that the regular character is $$b_1\chi_1+\dots+b_n\chi_n,$$ whether $b$ is an integer linear combination of the rows of $A$, or in other words whether the matrix equation $b=xA$ has a solution $x$ that is an integer vector. Luckily GAP has a function $\text{SolutionIntMat}$ that does this.

I don’t know whether there’s an easy way using only built-in functions to check whether there is a solution of $b=xA$ that is a non-negative integer vector, so this method might be missing some smaller examples that answer the original question.