$\newcommand\ku{\mathrm{ku}}$In Denis Nardin's lectures on stable homotopy theory, he reproduces Bruno Harris's proof of Bott periodicity from group completion and goes on to identify the Bott element generating $\tilde{\ku}(S^2)$ very explicitly as $H - 1$ with $H$ the tautological bundle on $\mathbb{C}P^1$. He also claims that Harris's map $B(BU \times \mathbb{Z}) \to U$ is given by multiplication by the element (link to the relevant point in the lecture) in the process of constructing periodic K theory from connective K theory.
However, this step is skipped in his lecture notes and it's noted as immediate in the lecture itself. I'm struggling to make the connection explicit and fill in this step. If one consults e.g., Hatcher's notes on vector bundles or Bott's original proof, the Bott map is constructed rather differently which makes it easier to unravel the relevant maps to see that they correspond to multiplication by $\beta$, but connecting Harris's constructions to these maps is a little non-obvious (and perhaps best done by showing that all of them are given by multiplication by $\beta$!)
Is there a more direct way to see that the following composite induced by multiplication by the Bott element $\beta = H - 1$ is equivalent to Harris's map $B(BU \times \mathbb{Z}) \to U$?
$$B(BU \times \mathbb{Z}) = \Omega^\infty \Sigma \ku \to \Omega^\infty (\Sigma^{-2} \ku \otimes \Sigma \ku) \to \Omega^\infty\Sigma^{-1} \ku = \Omega(BU \times \mathbb{Z}) \simeq U$$
