Why labour, capital, and output levels cannot be pinned down in perfect competition?
Thus, the furthest we can go in terms of characterising the equilibrium in this economy/firm relates to the optimal capital-labour ratio. In effect, nothing can be said about the level of inputs and outputs, $L^*$, $K^*$, or $Y^*$.


I don't think this is true. You combined two equations into one, and thereby lost information. The optimization problem is

\begin{equation}
 \max_{K,L} p\cdot AL^{\alpha}K^{\beta} - w \cdot L - r \cdot K
\end{equation}
Taking derivatives w.r.t. $L$ and $K$ yields two first order conditions and the only two unknowns are $L$ and $K$.
$$
\alpha \cdot p\cdot AL^{\alpha-1}K^{\beta} = w
$$
$$
\beta \cdot p\cdot AL^{\alpha}K^{\beta-1} = r.
$$
As you noted this implies the cost minimization equation of
$$
\frac{L^*}{K^*} = \frac{\alpha}{\beta}\frac{r}{w},
$$
but why go that way? You could get
$$
L = \left(\frac{r}{\beta \cdot p\cdot A K^{\beta-1}}\right)^{\frac{1}{\alpha}}
$$
and plug this into the first equation, yielding
$$
\alpha \cdot p\cdot A\left(\frac{r}{\beta \cdot p\cdot A K^{\beta-1}}\right)^{\frac{\alpha-1}{\alpha}}K^{\beta} = w.
$$
And then
$$
K^{\frac{\alpha\beta - (\alpha-1)(\beta-1)}{\alpha}} =
K^{\frac{\alpha + \beta - 1}{\alpha}}
= \frac{w}{\alpha \cdot p\cdot A\left(\frac{r}{\beta \cdot p\cdot A}\right)^{\frac{\alpha-1}{\alpha}}}.
$$
Clearly this is not very pretty but it is very solvable.

Note: I did not go into second order conditions but this is only a maximum if $\alpha + \beta &lt; 1$.