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[update] I moved the original answer to the bottom, although it was partially wrong - just to keep the reference.
Table of coefficients of p'th powers of a function $ \qquad \small f : f(x)^p = (1+a \cdot x+b \cdot x^2+c \cdot x^3 + \ldots )^p $

sdfsdfsdfsdfsdf

$$ \begin{array} {llll} x^0& \cdot \binom{p}{0}& \cdot (1) \\ +x^1& \cdot \binom{p}{1}& \cdot (1a) \\ +x^2& \cdot \binom{p}{1}& \cdot (1b) \\ &+\binom{p}{2}& \cdot (1a²) \\ & \\
+x^3& \cdot \binom{p}{1}& \cdot (1c) \\ &+\binom{p}{2}& \cdot (2ab) \\ &+\binom{p}{3}& \cdot (1a^3) \\ & \\ +x^4& \cdot \binom{p}{1}& \cdot (1d) \\ &+\binom{p}{2}& \cdot (2ac+1b^2) \\ &+\binom{p}{3}& \cdot (3a^2 b) \\ &+\binom{p}{4}& \cdot (1a4) \\ & \\ +x^5& \cdot \binom{p}{1}& \cdot (1e) \\ &+\binom{p}{2}& \cdot (2ad+2bc) \\ &+\binom{p}{3}& \cdot (3a^2c+3ab^2) \\ &+\binom{p}{4}& \cdot (4a^3b) \\ &+\binom{p}{5}& \cdot (1a5) \\ & \\ +x^6& \cdot \binom{p}{1}& \cdot (1f) \\ &+\binom{p}{2}& \cdot (2ae+2bd+1c^2) \\ &+\binom{p}{3}& \cdot (6abc+3a^2d+1b^3) \\ &+\binom{p}{4}& \cdot (4a^3c+6a^2b^2) \\ &+\binom{p}{5}& \cdot (5a4b) \\ &+\binom{p}{6}& \cdot (1a6) \\ & \end{array}

$$

Assumed that I got your function correct ( $ \small P(x) = \sum_{k=1}^{\infty} (-1)^{k-1} \cdot (k-1)! \cdot x^k $ ) I have polynomials for the coefficients of $ \small P^{\text{ o } h} (x) $ Here I assume, that your notation $ \small P^n$ means iteration and not power. (If that was wrong I can delete this answer) I constructed the matrix-logarithm of the matrix-operator $ \small M$ for the function $ \small P(x)$ and got by $ \small \exp(h \cdot \log(M)) $ the following powerseries, where the coefficients at x are polynomials in the iteration-parameter h :

$ \qquad \small \begin{array} {l} P^{\text{ o } h} (x) = & 1 \cdot x \\\ & - h \cdot x^2 \\\ & + (h^2 + h) \cdot x^3 \\\ & + (-h^3 - 5/2 \cdot h^2 - 5/2 \cdot h) \cdot x^4 \\\ & + ( h^4 + 13/3 \cdot h^3 + 9 \cdot h^2 + 29/3 \cdot h) \cdot x^5 \\\ & + (-h^5 - 77/12 \cdot h^4 - 125/6 \cdot h^3 - 511/12 \cdot h^2 - 295/6 \cdot h) \cdot x^6 \\\ & + O(x^8) \end{array} $

I was not yet able to construct the other function $ \small P_T(C)$ although I've tried with a certain vague idea; please show some of the coefficents $ \small b_n $ so that I can compare.