= Solution
Put $\mathcal G=G(M_\star+M_{\rm pl})$, $h=\sqrt{\mathcal G a(1-e^2)}$, $\theta=\omega+f$ and $L=n_{\rm pl}t$. Here $h$ is the <specific angular momentum>, $f$ is the <true anomaly> and $\omega$ is the planar <longitude of periapsis>. Differentiating the <Kepler orbit> equation with $\dot f=h/r^2$ gives
$$
\dot r=\frac{\mathcal G}{h}e\sin f=Ae\sin f,\qquad r\dot f=A(1+e\cos f).
$$
Resolve the <velocity> along the radial and tangential <unit vectors>, $\widehat{\boldsymbol r}=(\cos\theta,\sin\theta)$ and $\widehat{\boldsymbol\theta}=(-\sin\theta,\cos\theta)$. The identities $\sin f\cos\theta-\cos f\sin\theta=-\sin\omega$ and $\sin f\sin\theta+\cos f\cos\theta=\cos\omega$ yield
$$
\boxed{\dot x=-A(\sin\theta+e\sin\omega),\qquad \dot y=A(\cos\theta+e\cos\omega).}
$$
This is the <Kepler velocity hodograph>: the <velocity> runs around a circle whose centre is displaced by the <orbital eccentricity>.
The <centre of mass> condition gives the star's barycentric position $\boldsymbol r_\star=-\mu a_{\rm pl}(\cos L,\sin L)$. Translating to the star without rotating the axes gives $\boldsymbol r'=\boldsymbol r-\boldsymbol r_\star$ and $\boldsymbol v'=\boldsymbol v-\dot{\boldsymbol r}_\star$. From <Kepler's third law>, $n_{\rm pl}^2a_{\rm pl}^3=\mathcal G$, so
$$
\mu a_{\rm pl}n_{\rm pl}=A\kappa,\qquad \kappa=\mu\alpha^{-1/2}\sqrt{1-e^2}.
$$
Consequently
$$
\boxed{\dot x'=\dot x-A\kappa\sin L,\qquad \dot y'=\dot y+A\kappa\cos L.}
$$
The small displacement and the larger reflex <velocity> have different orders: $|\boldsymbol r'-\boldsymbol r|/a=\mu\alpha$, but the reflex speed divided by the circular speed is $\mu\alpha^{-1/2}$. These are the source of the <astrocentric osculating-element oscillations>.
For the coordinate ratio, the exact result is
$$
\frac{x'-x}{x}=\frac{\mu\alpha\cos L}{(r/a)\cos\theta}.
$$
For fixed $e<1$, $r/a\ge1-e$, so this is $O(\mu\alpha)$ on any angular region with $|\cos\theta|$ bounded away from zero. The printed coordinate claim is not uniform over a full orbit: at $x=0$ its ratio is undefined. The nonsingular statement is $|\boldsymbol r'-\boldsymbol r|/r\le\mu\alpha/(1-e)$, assuming the <pericentre distance> remains much larger than the binary separation.
Squaring the two components of the <velocity> gives
$$
\boxed{v'^2=A^2\left[1+e^2+2e\cos f+\kappa^2+2\kappa\{\cos(L-\theta)+e\cos(L-\omega)\}\right].}
$$
At $\mu=0$, $\kappa=0$ and $\mathcal G=GM_\star$. Since
$$
\frac{1+e^2+2e\cos f}{a(1-e^2)}=\frac2r-\frac1a,
$$
this reduces to the <vis-viva equation>, $v'^2=GM_\star(2/r-1/a)$.
Now let the barycentric <Kepler orbit> be circular, write $\Delta=L-\theta$ and $\varepsilon=\mu\alpha^{-1/2}\ll1$, and put $v_0=\sqrt{\mathcal G/a}$. The <osculating orbital elements> about the star use the gravitational parameter $GM_\star=(1-\mu)\mathcal G$. To the required order, $r'=a[1+O(\mu\alpha)]$ and $v'^2=v_0^2[1+2\varepsilon\cos\Delta+O(\varepsilon^2)]$. The <specific orbital energy> therefore implies
$$
\frac a{a'}=\frac{2a}{r'}-\frac{av'^2}{GM_\star}=1-2\varepsilon\cos\Delta+O(\mu+\varepsilon^2),
\qquad \boxed{\frac{a'}a-1=2\mu\alpha^{-1/2}\cos\Delta+O(\mu+\varepsilon^2).}
$$
Here $\mu=o(\varepsilon)$ and $\varepsilon^2=o(\varepsilon)$ under the stated hierarchy. Even at a phase where the leading cosine vanishes, the absolute error bound remains valid.
Finally, use the <eccentricity vector> $\boldsymbol e'=(\boldsymbol v'\times\boldsymbol h')/(GM_\star)-\widehat{\boldsymbol r}'$. In radial and tangential components it is
$$
e'_r=\frac{r'v_\theta'^2}{GM_\star}-1,\qquad e'_\theta=-\frac{r'v'_rv'_\theta}{GM_\star}.
$$
The translation changes the radial basis only by $O(\mu\alpha)$, whereas $v'_r=-v_0\varepsilon\sin\Delta+O(v_0\mu\alpha)$ and $v'_\theta=v_0(1+\varepsilon\cos\Delta)+O(v_0\mu\alpha)$. Thus
$$
\boxed{e'\cos(\omega'-\omega-f)=2\mu\alpha^{-1/2}\cos\Delta+O(\mu+\varepsilon^2),\qquad e'\sin(\omega'-\omega-f)=\mu\alpha^{-1/2}\sin\Delta+O(\mu+\varepsilon^2).}
$$
For the original circular <Kepler orbit>, $\omega$ and $f$ are separately arbitrary; their sum $\theta$ is well defined. Using the <eccentricity vector> avoids introducing an undefined <pericentre> for that circular <Kepler orbit>.
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