  Let $K = \mathbb{Q}(i)$. In this work, we study the Petersson inner product
of a Hermitian Eisenstein series of Siegel type on the unitary group
$U_{5}(K)$, diagonally-restricted on $U_2(K)\times U_2(K)\times U_1(K)$,
against two Hermitian cuspidal eigenforms $F, G$ of degree $2$ and an elliptic
cuspidal eigenform $h$ (seen as a Hermitian modular form of degree 1), all
having weight $k \equiv 0 \pmod 4$. This consideration gives an integral
representation of a certain Dirichlet series, which will then have an analytic
continuation and functional equation, due to the one of the Eisenstein series.
By taking $F$ to belong in the Maass space, we are able to show that the
Dirichlet series possesses an Euler product. Moreover, its $p-$factor for an
inert prime $p$ can be essentially identified with the twist by $h$ of a degree
six Euler factor attached to $G$ by Gritsenko. The question of whether the same
holds for the primes that split remains unanswered here, even though we make
considerable steps in that direction too. Our paper is inspired by work of
Heim, who considered a similar question in the case of Siegel modular forms.
