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This reminds a bit of Dustin Clausen's paper arxiv.org/abs/1110.5851. In particular the real J-homomorphism can be interpreted as a map of spectra $K(\mathbb{R}) \to \text{Pic}(\text{Sp})$. The author then constructs a "p-adic" version of this. Theorem 0.1 of cited document talks abou the image in the E(1)-local sphere.
Thanks Neil! I was being a bit lazy with my notation - everything is really at the prime 2. In this case $F(KO,L_{K(1)}S^0)$ is the fibre of the self-map $(\psi^3-1)_\ast$ of $F(KO,KO)$ so I guess this is the map to understand?
What about the remark in the end of the introduction of Mahowald-Rezk? Hu and Kriz (math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf) have constructed real oriented theories based on $E(n)$, which are usually denoted $ER(n)$. Now $ER(1) = KO_{(2)}$ and $ER(2)$ is very much like $TMF_0(3)$. They then make the comment "Presumably the construction of [HK01] can be carried out to construct a “real” version of $TMF_1(3)$"
I can also recommend Mark Behrens' talk recently at MSRI, which can be found msri.org/workshops/685. He explicitly shows the periodic families on Hatcher's charts
