As Jacob says, to produce a map $RP^\infty\to B\mathrm{Pic}(KU)$, we should use the fact that there is an infinite loop splitting $\mathrm{Pic}(KU)=\mathrm{Cliff}_{\mathbb{C}} \times Y$, where $Y$ is the 3-connected cover.
So, when I got to do the exercise of computing homotopy classes of maps $RP^\infty\to B\mathrm{Cliff}_{\mathbb{C}}$ (using knowledge of the $\infty$-loop k-invariants, as decribed in the other answers), I get $[RP^\infty, B\mathrm{Cliff}_{\mathbb{C}}] = \mathbb{Z}/8$: each stage in the postnikov tower of $B\mathrm{Cliff}$ adds a factor of $2$, and all the extensions are non-trivial. In particular, the elements which are congruent to $1$ mod 2 in $\mathbb{Z}/8$ are non-trivial on the bottom cell. So, if I've done the calculation right (and that's a big if), there are four distinct ways to make $\mathbb{Z}/2$ act so that the generator acts by suspension.
I find it interesting that I get $\mathbb{Z}/8$, which reminds me of real Bott periodicity. In fact, the calculations make it look like $$\mathrm{map}_*(RP^\infty, B\mathrm{Cliff}_{\mathbb{C}}) \approx \mathrm{Cliff}_{\mathbb{R}}$$ as infinite loop spaces.
Is there a way to take a real Clifford algebra $A$, and functorially produce from it a monoidal $\mathbb{Z}/2$-action on (the symmetric monodial Morita $2$-groupoid of) complex Clifford algebras?