Is it practical to derive the prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"? Is it practical to derive the optimal prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"?
Suppose you assume a probability distribution.
You know what the posterior is because you assume you know what the asymptotic convergences are.
Now you can derive the prior by multiplication.
 A: It is generally regarded as bad practice to decide the prior on the basis of the evidence.
It would often be possible to do as you suggest: take a desired posterior distribution, divide it by the likelihood of the evidence, and multiply the result by some number so you have a prior distribution which integrates to $1$ (i.e. is proper) which when recombined with the evidence would produce the posterior distribution (the calculations will be easier if the desired posterior is conjugate to the likelihood of the evidence). This looks like first deciding your conclusions and then adjusting your assumptions to reach your pre-decided conclusions.
There will be cases where this appears to produce apparently unusable results, in particular where the desired posterior distribution is more disperse than the evidence can justify.  This might even happen in your case where you assume an asymptotic result.
As an example, suppose you have a binomial experiment with unknown parameter $p \in [0,1]$ and your evidence is $2$ successes from $5$ attempts, so with likelihood $10p^2(1-p)^3$. If you want to have a uniform posterior distribution, i.e. with density $f(p)=1$ on $p \in [0,1]$, then your prior density would have to be proportional to $\frac1{10p^2(1-p)^3}$ and there is no constant of proportionality which would make this integrate to $1$, so your prior would be improper (in my view excessively so, even when compared to the Haldane prior).
A: Mathematically, starting from given densities $\pi(\theta|x)$ and $f(x|\theta)$, there is no reason for these two functions to be compatible, namely for$$\dfrac{\pi(\theta|x)}{f(x|\theta)}$$to factorise as a function of $x$ and a function of $\theta$.
