Using large-N technique at fixed dimension ($d=3$), I examine the multicritical behavior of a $U(N/2)\ifmmode\times\else\texttimes\fi{}U(N/2)$ Ginzburg-Landau theory of two multicomponent complex fields interacting through gauge fields described by Maxwell terms and a mixed Chern-Simons term. This model is relevant to the dynamics of Cooper pairs and vortices in a self-dual Josephson junction array system near its superconductor-insulator quantum transition. I present calculations of the various critical exponents including $1/N$ corrections to the $N=\ensuremath{\infty}$ saddle point. I investigate in the scaling region the behavior of the renormalized zero-momentum four-point quartic couplings $u$ and $w$ in the action, and I calculate the $1/N$ correction to the $\ensuremath{\beta}$-functions and their fixed-point values. It is shown that the decoupled fixed point is destabilized in the presence of the mixed Chern-Simons term at the next-to-leading order. Finally, I examine the universal character of the conductivity at the critical point up to the next-to-leading order in $1/N$ expansion.