HMMT 2022 A9

Suppose $P(x)$ is a monic polynomial of degree 2023 such that

$$P(k)=k^{2023} P\left(1-\frac{1}{k}\right)$$

for every positive integer $1 \leq k \leq 2023$. Then $P(-1)=\frac{a}{b}$, where $a$ and $b$ relatively prime integers. Compute the unique integer $0 \leq n<2027$ such that $b n-a$ is divisible by the prime 2027 .