= Solution
Use $\widetilde{\mathbb P}$ for the reference measure under which $X$ is a <Brownian motion>. Pathwise, $B_t=X_t-\int_0^tX_sds$. Put $h_s=X_s\mathbf1_{\{s\le T\}}$ and define
$$
Z_t=\exp\left(\int_0^{t\wedge T}X_s\,dX_s-\frac12\int_0^{t\wedge T}X_s^2ds\right).
$$
The stopped integrand has absolute value at most one. The <Novikov condition> holds on every finite horizon, so $Z$ is a density <martingale>. The <Itô formula> also gives the more useful identity
$$
\boxed{Z_t=\exp\left(\frac12X_{t\wedge T}^2-\frac12(t\wedge T)
-\frac12\int_0^{t\wedge T}X_s^2ds\right)\le e^{1/2}.}
$$
Thus $Z$ is <uniformly integrable> over the entire time axis, not just a true <martingale> on each finite horizon. Under the reference measure, $\widetilde{\mathbb E}(t\wedge T)=\widetilde{\mathbb E}X_{t\wedge T}^2\le1$, so $T<\infty$ almost surely. Hence $Z_t\to Z_T>0$ and $\widetilde{\mathbb E}Z_T=1$.
Define the <probability measure> on the original <sigma-algebra> by
$$
\boxed{\frac{d\mathbb P}{d\widetilde{\mathbb P}}=Z_T.}
$$
Its restriction to $\mathcal F_t$ has density $Z_t$. The <Girsanov theorem> states that subtracting the integrated density integrand from the reference <Brownian motion> gives a <Brownian motion> under the new measure. Therefore
$$
W_t^{\mathbb P}=X_t-\int_0^{t\wedge T}X_sds
$$
is a <Brownian motion> under $\mathbb P$, and $B_{t\wedge T}=W^{\mathbb P}_{t\wedge T}$. This is exactly the requested Brownian property until $T$. The <bounded Girsanov density for exit of an unstable linear diffusion> also proves global absolute continuity, so a mere collection of unspecified finite-horizon measures is unnecessary.
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