Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles

An evident general construction is to take any multiplicative cohomology theory $E^*$ and a cohomology operation $\psi$ in that theory, say taking $E^q$ to $E^{q+n}$. For an $E$-oriented real $q$-bundle $\xi$ over $X$, $\theta^{-1}(\psi(\mu))$ gives the value of a characteristic class, where $\mu$ is the Thom class of $\xi$ and $\theta$ is the Thom isomorphism. For stable $\psi$, defined for all $q$, this will give a characteristic class on all $E^*$-oriented bundles of the sort wanted, modulo precision about condition (ii). [Senior moment nonsense eliminated]. You are studying the J-map $BO\to BF$ (or $BU \to BSF$), where $BF$ classifies stable spherical fibrations (oriented for $BSF$). A lot more is known than Adams knew. In particular, he didn't yet have the Adams conjecture. Rationally, $BF$ is a point. At an odd prime $p$, $BF$ factors as $BJ\times BCokerJ$, and at $2$ there is a non-split fiber sequence $BCokerJ \to BSF \to BJ$. The $J$-map at $p$ is best thought of as a map $BSpin\to BSF$ ($BO\simeq BSpin$ at $p>2$). By the Atiyah-Bott-Shapiro orientation, the $J$-map $BSpin\to BSF$ factors through the classifying space $B(SF;kO)$ for $kO$-oriented spherical fibrations and, at any $p$, $B(SF;kO)$ splits as $BSpin\times BCokerJ$ (but BSpin is seen in two pieces, one carrying the Wu classes, the other the rest of the Adams splitting). The intuition is that $BCokerJ$ and thus the unknown parts of the stable homotopy groups of spheres can be ignored, leaving the focus on the quite computable composite $BSpin \to BSF \to BJ$. This is too fast, and details are in Chapter V of $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra''. In ordinary mod $p$ cohomology, calculations are thoroughly understood but don't shed light on your question. They are also understood for $K$-theory, by work of Hodgkin and Snaith, and here the intuition that $Coker J$ can be ignored is made precise by Hodgkin's result that $\tilde K(BCokerJ)=0$.

You might have luck with the "universal" choice of $h$, namely the cohomology theory given by the connective spectrum corresponding to the $E_{\infty}$-space of stable spheres under smash product (as a plain space, this is $\mathbb{Z}\times B\operatorname{Aut}(S)$). The characteristic class map $KO \to h$ you want is then essentially the real $J$-homomorphism, studied extensively by Adams in a series of papers.

This $h$ isn't very computable, though, since for instance its coefficient groups in degrees $> 1$ are the stable homotopy groups of spheres in degrees $> 0$. For a more computable variant I would suggest working completed a prime $p$ and taking $h'$ = suspension of the $K(1)$-local sphere = suspension of fiber of $\psi^u - 1 : KO \to KO$ for $u$ a generator of $\mathbb{Z}_p^*/\pm 1$, where the map $h \to h'$ is gotten from Rezk's logarithm in the $K(1)$-local case. Adams's results more-or-less imply that this $h'$ detects a lot of the same information as $h$ (on coefficient groups, the map $h \to h'$ is Adams's $e$-invariant, up to normalization), but $h'$ is a much more tractable cohomology theory.

Edit: Oh, I should say, up to a unit factor (at least at an odd prime), the composite map $KO \to h \to h'$ identifies with the boundary map in the fiber sequence $L_{K(1)}S \to KO \to KO$. So it's very computable.