Let
$$ X \sim \text{Poisson}(\lambda) \\ p = P(X \lt k | X \ne k; \lambda) \\ p = \frac{\text{CDF}(k - 1; \lambda)}{1 - \text{PMF}(k; \lambda)} $$
Questions:
- Does $p$ - i.e. in words the probability that $X$ is less than $k$ given it is not equal to $k$ - have some specific name?
- Is there an efficient way to calculate $\lambda$ given $k$ and $p$? As opposed to simple numerical optimisation
(1) Not to my knowledge
(2) Since you know your variable is Poisson, you know the expression for the CDF and PMF. You then solve for $\lambda$ in the following formula:
$$p= F(k-1) = \frac{1-e^{-\lambda (k-1)}}{\left(\tfrac{e^{-\lambda}\lambda^k}{k!}\right)}.$$
This fried Wolfram Alpha, but I’m sure that some root-finding algorithm will help you. Since you explicitly say thay you are not interested in numerical solvers, from my knowledge there are no closed-form expressions for $\lambda$ in terms of well-known functions