  In this paper, we compute certain $p$-adic zeta integrals appearing in the
doubling method of Garrett and Piatetski-Shapiro-Rallis for unitary groups.
Using structure theorems in the author's work arXiv:2310.09110 for
$P$-(anti-)ordinary automorphic representations involving Bushnell-Kutzko
types, we associate local Siegel-Weil sections at $p$ to such Bushnell-Kutzko
types. Then, fixing compatible choices of $P$-anti-ordinary vectors, we find
explicit formulae relating the corresponding $p$-adic zeta integral to modified
$p$-Euler factors and volumes of $P$-Iwahoric subgroups. Our results extend the
ones of Eischen, Harris, Li and Skinner for the ordinary setting by allowing
automorphic representations with nontrivial supercuspidal support at $p$.
