I have a set of observations that I'm assuming are the sum of two binomials ie. $Y_t=X_{1,t}+X_{2,t}$ where $X_{1,t}\sim Bin(N_{1,t},p_1)$ and $X_{2,t}\sim Bin(N_{2,t},p_2)$ $t\in\mathbb{N}_0$. After some time $T$, $N_{1,t}=N_1$ and $N_{2,t}=N_2$ $\forall t$ such that $t>T$. The process generating $X_{1,t}$ and $X_{2,t}$ is a chain block model and the $X_{1,t}$ does not retain anything from the previous time step ie: all leave every time step.
The only knowledge I have is that $N_2 \gg N_1$ and $p_2\ll p_1$. Consequently, $Y_t$ will be comprised mostly of $X_{1,t}$ and $\mathbb{E}(X_{1,j})=O(10)$ and $\mathbb{E}(X_{2,j})=O(1)$. I'd like to estimate $p_1$ and maybe $p_2$.
I've tried maximum likelihood estimation to estimate $N_{1}$ and $p_1$, by considering the process $X_{2,t}$ to be noise, and got an answer. I've considered parameter estimation using counting processes, but there is a lot I need to learn before I can understand the feasibility for this problem and implement it. There is also the possibility of mle using the saddlepoint approximation, but most of the papers concerning this are very recent with multiple different ways to implement it. I'm looking for a recommendation for an approach.