Let $a,b,c,d$ be positive integers, with $\gcd(c,d)=1$, such that $$a^2+3b^2=cd.$$ By well-known classical results, we have that $c$ and $d$ are both of the form $u^2+3v^2$.
QUESTION: Is it valid to claim that that there exists at least one representation \begin{align} (c,d) = (p^2+3q^2,\ r^2+3s^2) \end{align} in [not necessarily positive] integers $p,q,r,s$ such that \begin{align} (a,b) &= (pr+3qs,\ ps-qr) &\text{or}&& (a,b) &= (pr-3qs,\ ps+qr), \end{align} or does this commit some sort of unique factorization error?
yes, this is Theorem 69 on page 91 of Dickson, Introduction to the Theory of Numbers (1929). You have made it easier by demanding $c,d$ coprime. The reason this matters is that Dickson's forms need not be primitive, so $2x^2 + 2 xy + 2 y^2$ might come up