Some preliminary remarks. Lukacs's paper does not refer to or mention Pourchet, Prouhet, or any similarly spelled name. Johnson gives a detailed history of Faa di Bruno's formula. He does not mention such a name either, but comments "one is struck by the obscurity of many of the names in our story... It would be nice to have more biographical information about some of them..."
I have a feeling that the only way to confirm would be to find Comtet's formula in Prouhet's papers. Comtet's book (p.331) references one Prouhet's paper in particular, in Nouvelles Annales M[athématiques], 5 (1866) 384, and nothing by "Pourchet". Unfortunately, Numdam shows no Prouhet's paper in 1866 Nouvelles Annales, on p.384 or elsewhere, and the one from 1867 has nothing of the sort in it. Then again Numdam does not show his 1864 and 1865 papers, which appear on Gallica-Math list. Both agree on nothing from Prouhet in 1866 though.
Even if it is there he may not have been the first to derive the formula. Faa di Bruno published 'his' formula in December of 1855, but "that several other mathematicians found different expressions for the m-th derivative of $g(f(t))$ in the 19th century has been forgotten; these are all independent of Faa di Bruno's work and a few of them predate it. Most of all, Faa di Bruno was neither the first to state the formula that bears his name, nor the first to prove it... Other mathematical accomplishments... have caused him to receive more credit for the higher chain rule than he deserves..."
The first proof appeared in volume 3 of Lacroix's Traite du Calcul, "a snapshot (or perhaps a mural) of 18th century analysis", 2nd edition only, which came out in 1819. His source appears to be 1800 Du Calcul des De'rivations of Arbogast, who "Arbogast gives a prose rule for writing down the general case, and on p. 313 he illustrates his rule by writing down the case n = 6 again, twice, followed by n = 7. Although he never gives a general formula, nor a proof of his rule... there is a formula for the coefficient of $x^m$ in $(\zeta + \gamma x + \delta x^2 + \varepsilon x^3 + \text{etc.})^n$. Thus Arbogast had most of the ingredients of Lacroix's argument at hand, but he seems never to have written down Faa di Bruno's formula as such.".