Stability of the leading Laurent term under a small perturbation for polynomial coordinates of $\mathbb A^2$

Suppose that $$ x=u^{-d},\qquad y=f(u), $$ where $$ d\in \mathbb Z_{>0},\qquad f(u)\in \mathbb C((u)), \qquad y=o(x). $$ Equivalently, $\operatorname{ord}_u f(u)>-d$.

Let $ (P,Q)\in \operatorname{Aut}_{\mathbb C}\mathbb C[x,y]$ be a polynomial coordinate system, and write $$ X=P(x,y),\qquad Y=Q(x,y). $$ Assume also $$ Y=o(X), $$ this implies $\operatorname{ord} X<0$, put it $-D$.

I want to prove the following statement.

For every $ \Delta\in u^d\mathbb C[[u]] $ and for an auxiliary parameter $\alpha$, one has $$ P(x,y+\alpha\Delta)=c u^{-D}+O(u^{-D+1}) \qquad c\in\mathbb C $$ Equivalently, the coefficient of the lowest $u$-order term of $P(x,y+\alpha\Delta)$ is independent of $\alpha$.

A Taylor expansion gives $$ P(x,y+\alpha\Delta) =P(x,y) + \sum_{k\ge 1} \frac{1}{k!} \partial_y^kP(x,y)\alpha^k\Delta^k. $$ Thus it would be enough to prove $ \operatorname{ord} \partial_y^kP(x,y)+kd>-D $ for all $k\ge 1$, in the assumptions described above.

Please help me with this problem.