What am I assuming when I make this approximation?
The approximations you can do are the following :

1 - Practically all silver atoms are included in the complex ion. So that the concentration of that ion is equal to [Br-]

2 - The amount of AgBr dissolved is negligible with respect to the amount of Na2S2O3 available. So that, at the end, $\ce{[S2O3^{2-}]}$ is equal to the concentration M of the thiosulfate.

With these approximations, you may rewrite $\beta{_2}$, so that [Ag+] is :
$$\ce{[Ag+]= {[Ag{_2}(S_2O_3^{3-}]}/{(M·{\beta_2})}} = [Br^-]/(M·{\beta_2})$$ 
With the solubility product of AgBr , you have a system of two equations with two unknowns. The first solution [Ag+] will be very small. But the second solution, namely [Br-] gives you the solubility of AgBr you are looking for.