Splitting of $BGL_1(KR)$

Question

There are infinite loop space splittings $BGL_1(KO)\simeq BGL_1(KO)[0,2]\times Z$ and $BGL_1(KU)\simeq BGL_1(KU)[0,3]\times Z'$ where $Z$ and $Z'$ are 2 and 3 connected, respectively (i.e. they have trivial homotopy groups up through degree 2 and 3). Is there a $ℤ/2$-equivariant space $BGL_1(KR)$ which has an equivariantly truncated split summand on the bottom whose underlying space is $BGL_1(KU)[0,3]$ and whose fixed points are $BGL_1(KO)[0,2]$? If so, where should one truncate it? I don't have a lot of familiarity with equivariant truncations but I imagine this involves trivializing maps out of some set of representation spheres? Here $KR$ is so-called real $K$-theory.