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The paper [https://archives.maths.anu.edu.au/people/Kovacs/K033.pdf] by Bryant and Kovacs is very relevant here, at a reasonably generic level. You have to be careful about the action of the centre of ${\rm SL}_{n}(K)$: note that the centre has order $d = {\rm gcd}(n,|K|-1).$ The result of Bryant and Kovacs shows that if $m$ is large enough, then the regular module is a direct summand of $V^{m} \oplus V^{m+1} \oplus V^{m + d-1}$, where I let $V^{j}$ denote the $j$-fold tensor product of $V$ with itself. Note that each irreducible representation restricts to a multiple of a one-dimensional representation of the centre, and that for a given choice of one-dimensional representation $\sigma$ of $Z({\rm SL}(n,K))$, the irreducibe rpresentations (of ${\rm SL})$ which lie over $\sigma$ occur in one and only one of the $d$ successive tensor powers of $V$.