Asymptotic behavior of the ratio between the largest two singular values of product of i.i.d. random complex matrices

So this is correct. The theorem that you need is the multiplicative ergodic theorem. Expressing it in your language, it states that $\frac 1n\log s_i(A_n)\to\lambda_i$, where $s_i$ is the $i$th singular value of the matrix product; and $\lambda_i$ is the $i$th Lyapunov exponent of the system. In order to have what you want, you require that $\lambda_1$, the largest Lyapunov exponent of the system, is strictly larger than $\lambda_2$, the second largest Lyapunov exponent. This is guaranteed by a theorem of Guivarc'h and Raugi ("Products of Random Matrices: Convergence Theorems") that applies in any of the models for the randomness that you are assuming. That paper gives criteria that guarantee that the top Lyapunov exponent is simple (which is exactly what you want). Later papers due to Gol'dsheid and Margulis gave criteria for simplicity of the entire Lyapunov spectrum.