Much about your question is unclear, but I can talk about the median in the gamma distribution and you may be able to resolve the problem from that.
The gamma distribution is typically written either in the rate form or the scale form (I'll avoid using $\beta$ here, since your intention isn't clear):
Rate form:
$$f(x;\alpha,\phi) = \frac{\phi^\alpha}{\Gamma(\alpha)} x^{\alpha-1}e^{-\phi x} \quad \text{ for } x > 0 \text{ and } \alpha, \phi > 0$$
Scale form:
$$f(x;\alpha,\theta) = \frac{x^{\alpha-1}e^{-\frac{x}{\theta}}}{\theta^\alpha\Gamma(\alpha)} \quad \text{ for } x > 0 \text{ and } \alpha, \theta > 0$$
It's sometimes also written in the mean form (especially for GLMs).
The mean is $\alpha\theta = \alpha/\phi$.
The median is approximately $\frac{3 \alpha - 0.8}{3 \alpha + 0.2}$ times the mean (as long as $\alpha$ is not too small).
I think I know what an appropriate β parameter is (though I could be wrong on this point, feel free to make comments if I'm thinking about that incorrectly)
It's a bit hard to comment about whether you have a misconception here if you aren't explicit about what you actually think and why!