I'm having some trouble following your thinking, but the simple truth is that, if $ x^2 + y^2 + z^2 = 4n,$ then $x,y,z$ must all be even, so we get an integral expression $ \left( \frac{x}{2} \right)^2 + \left( \frac{y}{2} \right)^2 + \left( \frac{z}{2} \right)^2 = n. $ That is, $n$ is the sum of three squares if and only if $4n$ is the sum of three squares. Furthermore, among numbers that are not divisible by $4,$ those that are $1,2,3,5,6 \pmod 8$ are expressible as the sum of three squares, while those that are $7 \pmod 8$ are not. The difficult theorem is the part about $1,2,3,5,6 \pmod 8$ succeeding.
So the way I would word the thing, keep dividing out by $4$ until the number is no longer divisible by $4.$ Check that number to see whether you get $1,2,3,5,6 \pmod 8$ or $7 \pmod 8.$
Maybe this example will show something: instead of 136 take 240, as in $240=4^0[8(30)+0] = 4^1[8(7)+4] = 4^2[8(1)+7]. $ Either of the expressions with $4^0$ or $4^1$ gives you a misleading impression, since $240$ is not the sum of three squares.