Suppose I am deriving a length contraction formula using natural units. If I arrive at $L = L_0 \sqrt{1 - v^2}$, I know that I should divide $v^2$ by $c^2$ to get the correct answer in SI units. But what if I mistakenly forgot to square the velocity and arrived at $L = L_0 \sqrt{1 - v}$. I would then be inclined to divide $v$ by $c $ and conclude that the answer is $L = L_0 \sqrt{1 - \frac{v}{c}}$.

If I had not used SI units during derivation and only forgot to square the velocity, I would have arrived at $L = L_0 \sqrt{1 - \frac{v}{c^2}}$. I could have kept track of the dimensions and told myself I had made a mistake somewhere. Is this the disadvantage of natural units? Or is there a way to get around this problem?