Axioms and uniqueness for the Euler class

Question

In this question it was asked if the 4 properties listed on the wikipedia page uniquely characterise the Euler class. I answered no and claimed:

For every oriented vector bundle $E\to X$ of rank $n$ there exists a unique class $e(E) \in H^n(X;\mathbb{Z})$ satisfying the following axioms:

1. Naturality: For every $f:Y \to X$ we have $e(f^*E) = f^*e(E)$.
2. Sum formula: If $W\to X$ is another oriented vector bundle we have $e(W\oplus V) = e(V)\cup e(W).$
3. Orientation: If $\bar{E}$ is $E$ with the opposite orientation, we have $e(\bar{E}) = -e(E)$.
4. Normalisation: For the (real oriented) tautological bundle $\gamma^1 \to \mathbb{CP}^1$ we have $<e(\gamma_1),\;[\mathbb{CP^1}]> = -1$.

I thought one could prove this using a real oriented splitting principle, that is for $E \to X$ we construct a map $f:Y \to X$ with \begin{align*}f^*E \overset{\sim}{=} P_1 \oplus ... \oplus P_k \oplus \xi \end{align*} where the $P_i$ are oriented 2-plane bundles and $\xi = \underline{\mathbb{R}}$ and $k=(n-1)/2$ if $n$ is odd and $\xi = 0$ and $k=n/2$ if n is even.

But Jack Lee pointed at some trouble. If $f^*:H^*(X) \to H^*(Y)$ is injective, I would be fine. But in the $n$ odd case, this cannot be true, since $e(f^*E) = 0$ while $e(E)$ can be non-trivial two torsion. In the even case there is Proposition III.11.2 of Spin Geometry by Lawson & Michelsohn which claims that $f^*$ is injective but their proof is not clear ( see this question by Jack Lee).

So my question is, if anybody can give a proof of the uniqueness statement (or acounterexample, in the case they are not uniquely characterising the class)?

Any proof is welcome, but my goal was to stay as close as possible to the corresponding proofs for the uniqueness of Stiefel-Whitney and Chern classes via splitting principles. Even without the injectivity of the pullback this could be doable.