Here's a method that gets close. Do a continued fraction expansion of $\frac{m}{n}$:
$$\frac{m}{n} = \frac{1}{a + \frac{1}{b + \epsilon}} \approx \frac{1}{a + \frac{1}{b}} = \frac{1}{a} - \frac{1}{a(1 + ba)}$$
If $\epsilon$ is small, then this is probably very close to optimal if not optimal in most cases.
The method fails when the truncated continued fraction approximation isn't very good. Take $\frac{15403}{26685}$ (a rational approximation to the Euler-Mascheroni constant) as an example. The continued fraction approximation is:
$$\frac{15403}{26685} = \frac{1}{1 + \frac{1}{1 + \epsilon}}$$
Which suggests:
$$\frac{15403}{26685} \approx \frac{1}{1} - \frac{1}{2}$$
But clearly $\frac{1}{2} + \frac{1}{14}$ is closer.
I would wager that the worst case for any method is the conjugate golden ratio $\phi' = \frac{\sqrt{5} - 1}{2}$. This isn't rational (in fact it's the "most" irrational number in a technical sense; find its continued fraction expansion if you're curious), but you can get arbitrarily close by choosing $m$ and $n$ to be consecutive Fibonacci numbers, e.g. $\frac{4181}{6765}$. The best approximation I could find is $\frac{1}{2} + \frac{1}{9}$.