Suppose that we have a sequence of observations $y_1, y_2, ... y_n$ that we know is generated by $ax_1, ax_2, ...ax_n$ where $X \sim \mathrm{Poisson}$. Can we use MLE to estimate $a$ and the $\lambda$ of the Poisson distribution?
EDIT: I deleted my attempt at the MLE, it was not helpful.
Your likelihood is not correct, so everything you've computed makes no sense.
If $Y_i = aX_i$ where each $X_i \sim \operatorname{Poisson}(\lambda)$, then $\Pr[Y_i = y] = \Pr[X_i = y/a]$, with the obvious ramification that if $y/a$ is not a nonnegative integer, the probability is zero. Thus the joint likelihood is $$\mathcal L(a, \lambda \mid \boldsymbol y) = \prod_{i=1}^n e^{-\lambda} \frac{\lambda^{y_i/a}}{(y_i/a)!} \mathbb 1 (y_i/a \in \mathbb Z^+ \cup \{0\}).$$
From a practical perspective, if $a$ could be any real number, given a sufficiently large $n$ we can exploit the integer support of $X$ to estimate $a$ with great precision. If $a$ is itself an integer, we simply look for the least common divisor of the observations.