A preprint by Stefan Schröer came out today with the answer to this question: arXiv:2004.07025.
No such Enriques surface exists. In fact, there is no classical Enriques surface over $\mathbb F_2$ with 25 $\mathbb F_2$-points (and with the extension of $\mathbb Z^{10}$ by $\mathbb Z/2$ split in the Picard group, which you can also deduce).
