$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-spaces of invertible $KO$ and $KU$ modules respectively. I am interested in understanding the 2 and 3-truncations, respectively, of these spectra. We know that the underlying spaces of these truncations split, i.e. $$\Pic(KO)[0,2]=\Omega^\infty \pic(KO)[0,2]\simeq \mathbb{Z}/8\times K(\mathbb{Z}/2,1)\times K(\mathbb{Z}/2,2)$$ and $$\Pic(KU)[0,3]=\Omega^\infty \pic(KU)[0,3]\simeq \mathbb{Z}/2\times K(\mathbb{Z}/2,1)\times K(\mathbb{Z},3).$$
It seems to be "well known" that the first two $k$-invariants of $\pic(KO)$ are $Sq^2\circ\rho\colon H\mathbb{Z}/8\to\Sigma^2H\mathbb{Z}/2$, where $\rho$ is reduction mod 2, and then what is essentially another $Sq^2$ (the actual $k$-invariant ignores the $\mathbb{Z}/8$). Similarly, it seems to be "well known" that the first two $k$-invariants of $\pic(KU)$ are a $Sq^2$ and $\beta\circ Sq^2$ (again, this second one is a bit fudged because it ignores the $\pi_0$ part).
This fact about $\pic(KU)$ is stated in the proof of Proposition 7.14 of this paper of Gepner and Lawson, but a reference is not given. I also believe I know an argument for, at the very least, the second $k$-invariant of $\pic(KO)$. Indeed, one can compute the possible $k$-invariants that each of the above can be and show that if they're non-trivial then they're the ones I've described. But all of this seems like it must be written down somewhere already, and I'd prefer not to reinvent the wheel, if possible.
So that's the question, does anyone know of any concrete proofs of these facts about the first two $k$-invariants of $\pic(KO)$ and $\pic(KU)$?
