Let $A = \text{diag}(a)$, for $a\gt 0$ in $\Bbb{Z}$, be an $n\times n$ matrix. I want to factor it into a matrix $B$ such that $B^e = A$ for fixed positive integer $e$.
All entries in $B$ are non-negative. Also, assume we can be certain that the entries in $B$ will not exceed $3$.
Is there a way to do this?
No. Consider $n = 1$, $e = 2$, and $a = 25$. Then the only possible (single) entry for $B$ is $5$, which is not less than $3$.
Perhaps your second paragraph means something different from the way I've interpreted it; if so, you need to get your quantifiers figured out.