Bounding Poisson CDF by the Binomial: $\Pr[B(n,p)\le a] \ge \Pr[P(np)\le a]$ when $np\le a$ This question is related to Poisson CDF as lower bound to binomial CDF .
That is, I seek to prove the inequality
$$\Pr[X\le a] \ge \Pr[Y\le a],$$
where $X\sim \text{Binomial}(n,p)$ and $Y\sim \text{Poisson}(np)$.
The difference is that I add the condition $np\le a$, which from numerical experiments seems to be exactly what is needed, as seen in the plot below:

For instance, this implies the inequality
$$\Pr[X\le \lceil np\rceil] \ge \Pr[Y\le \lceil np\rceil],$$
which is likewise easy to test numerically for reasonable ranges of $n$ and $p$.
The case $a=0$ is trivial, since in this instance we must have $p=0$ or $n=0$ and both sides are 1.
The $a=1$ case is more tricky, but I can prove
$$(1-p)^n + np(1-p)^{n-1} \ge e^{-np} + np e^{-np}$$
when $np\le 1$ using analytical methods.
The general case seems untractable to me, however.
I wonder if this coupling is well known?
Perhaps there is a probabilistic argument? Maybe something using characteristic functions?

[Not an answer -- sharing some thoughts that didn't fit in a comment. Please let me know if this is the wrong way of doing it.]
Have you thought about the problem as sum of iid RVs?
Let $X_1, \ldots, X_n$ be iid RVs where $X_i \sim \text{Binomial}(1,p)$. Also, let $Y_1, \ldots, Y_n$ be iid RVs where $Y_i \sim \text{Poisson}(p)$. Then, let $X^{(n)} = \sum_{i \in [n]} X_i$ and $Y^{(n)} = \sum_{i \in [n]} Y_i$. In this case $X^{(n)}$ and $Y^{(n)}$ are identically distributed as $X$ and $Y$ in the original problem.
A few observations:
(a) $n=1$ is easy, i.e., $P[X^{(1)} \leq a] \geq P[Y^{(1)} \leq a]$ for any $a \geq p.$ Perhaps some sort of induction on $n$ could be done?
(b) Let $M_{X_i}(t) = (1-p) + pe^t$ and $M_{Y_i}(t) = e^{p(e^t-1)}$ be the MGFs of $X_i$ and $Y_i$ respectively. Then, for all $t,$ $M_{X_i}(t) \leq M_{Y_i}(t)$ (direct consequence of $e^x \geq 1+x$). This also implies that $M_{X^{(n)}}(t) \leq M_{Y^{(n)}}(t)$ for all $n$ and $t$. Does this help to prove the claim?
