Not exactly what you want, but pretty close. I might have gotten the groupings wrong, but try something like.
MouseNew %>%
tidyr::unite("Behavior_Treatment",c(Behavior, Treatment), remove=FALSE) %>%
ggplot(aes(x = Behavior_Treatment, y = AverageExpression, fill=protein)) +
geom_col(position="dodge")+
facet_wrap(~Genotype, ncol=2) +
theme_classic()
To explain a bit, I created a new variable with unite() which is just concatenating Behaviour and Treatment. Use the new variable to make the bar plot.