= Solution
For a uniform distribution of phases along one period, average the ensemble <force> $BF_{\rm bound}$:
$$
\langle F\rangle=\frac{G_0}{P}\left[
\frac1{x_1}\int_0^{x_1}B(x)\,dx-
\frac1{P-x_1}\int_{x_1}^P B(x)\,dx\right].
$$
Using the negligible-carry-over form in <4/b> gives
$$
\int_0^{x_1}B\,dx=B_0\left[x_1-\frac{1-e^{-\alpha x_1}}\alpha\right],\qquad
\int_{x_1}^PB\,dx=\frac{B_0(1-e^{-\alpha x_1})}{\beta}
\left[1-e^{-\beta(P-x_1)}\right].
$$
Consequently
$$
\boxed{\langle F\rangle\simeq\frac{G_0B_0}{P}\left[
1-\frac{1-e^{-\alpha x_1}}{\alpha x_1}
-\frac{1-e^{-\alpha x_1}}{\beta(P-x_1)}
\left(1-e^{-\beta(P-x_1)}\right)\right]},
$$
where $\boxed{B_0=k_1/(k_1+k_2),\ \alpha=(k_1+k_2)/v,\ \beta=k_2/v}$. This is the printed expression to the accuracy justified by the velocity condition.
It is not an exact identity for the periodic problem at arbitrary $v$. Keeping $b_{\rm in}$ instead gives
$$
\langle F\rangle_{\rm exact}=\frac{G_0B_0}{P}\left[
1-\frac{(1-a)(1-q)}{(1-aq)\alpha x_1}
-\frac{(1-a)(1-q)}{(1-aq)\beta(P-x_1)}\right].
$$
For a concrete counterexample outside the small-velocity approximation, choose $k_1=k_2=P=G_0=v=1$ and $x_1=1/2$. Then the exact periodic value is about $+0.01976$, whereas the printed reset expression is about $-0.06478$. The discrepancy is due to nonzero incoming occupancy, not an integration sign change.
As $v\to0^+$, both $\alpha$ and $\beta$ diverge, and both formulas give
$$
\boxed{\lim_{v\to0^+}\langle F\rangle=\frac{G_0}{P}\frac{k_1}{k_1+k_2}}.
$$
The bound population occupies the downhill binding region, while occupancy on the uphill region is confined to a vanishing boundary layer. A nonzero limiting stall <force> is compatible with zero mechanical power $v\langle F\rangle$. The spatially restricted binding and uniform unbinding do not obey local <detailed balance> for this sawtooth <free energy>: a passive equilibrium model would require $k_1(x)/k_2(x)$ to reflect the local energy difference and not vanish on an interval of finite energy. This is a driven or prescribed-kinetics motor model; the equilibrium relation in <4/a> cannot be imposed on it without an energy supply and additional states.
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