Is #PE (#P Easy) closed under decrement?
If $\text{#PE}$ is closed under decrement then $\text P=\text{NP}$: consider $\text{#SAT}+1$. Conversely if $\text P=\text{NP}$ then $\text{#PE}=\text{#P}$ is closed under decrement because we can always find an accepting path if one exists then ignore it.

I don't know whether $f\in\text{#PE}$ always implies $f_{-1}\in\text{#P}$. Your $M$ might not provide an accepting path for us to ignore.