Could you explain how you found that example form? I tried to write some Sage code to search for these forms by generating a basis and then reducing mod p and solving a matrix equation, but when I ran it on weight $9/2$, $\Gamma_1(80)$ (Sage requires $16|N$ for some reason), it didn't find the form you mentioned, so I must be doing something wrong.

Thank you very much for the answer! In the Gross paper the result you state is proven in proposition $4.9$, by showing that the kernel of reduction mod $p$ is the principal ideal $(1-A)$, where $A$ is the Hasse invariant. The author does indeed assume $p$ does not divide $N$ generically throughout section $4$, but determining whether or not this assumption is actually needed for proposition $4.9$ is well beyond my capabilities.

@GTA Naively it seems to me as though the only way to get a form this way which is constant mod $p$ would be to bracket $E_{p-1}$ with a half integral weight form that is already constant mod $p$, which isn't helpful. I only need a single form for each prime.

@Kimball I had not heard of Cohen-Eisenstein series before, thank you for bringing them to my attention. From the definition it's not clear to me how they might be useful. Would you mind elaborating a bit on your thoughts?