Rationality of quaternionic Eisenstein series on $\mathrm{U}(2,n)$

Let $\mathbf{G}=\mathrm{U}(2,n)$ be the unitary group associated to a Hermitian space over a quadratic imaginary number field $E$. We assume that 2 is unramified in $E$, and the Hermitian space splits at all finite places and has signature $(2,n)$, where $n\equiv 2 \operatorname{mod} 4$. A theory of Fourier expansions of quaternionic modular forms on $\mathbf{G}$ is developed by Hilado, McGlade, and Yan. In this paper, we define a family of degenerate Heisenberg Eisenstein series $E_{\ell}$ for $\ell>n$ on $\mathbf{G}$, which is a weight $\ell$ quaternionic modular form, and we explicitly compute their Fourier expansions. We prove that the Fourier coefficients of $E_{\ell}$ are rational in a certain sense, and that their denominators are uniformly bounded by an integer depending only on $\ell,n$, and $E$. This provides the first family of quaternionic Eisenstein series whose Fourier coefficients are known to be rational or algebraic.