We can attack the baby problem directly, as Jairo Bochi and I did jointly in the comments.
For instance, the distribution which is a mixture of:
- $3/5$ of a Bernoulli distribution with $p=1/2$
- $2/5$ of a Beta distribution with $\alpha=\beta=4$
yields an expectation equal to $503582/8204625$, or approximately $.0614$. This mixture of a discrete and a continuous distribution seems to be almost optimal.