How do you carry errors for brightness magnitudes
You need to do what is called “propagation of uncertainties”. You can search to get more information on that, but briefly if you have some function $f(x)$ that depends on variable $x$, then the uncertainty $\sigma_x$ on the quantity $x$ is related to the uncertainty on $f$ by
$$
\sigma_f = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 \sigma_x^2}
$$
Here $f$ is your relative magnitude $m_1 - m_2$, and $x$ is your flux ratio.  So taking that partial derivative will give you the factor that relates the two uncertainties.