= Solution
Write $x=x_0+y$, so the fluctuation has homogeneous <Dirichlet boundary conditions> $y(0)=y(T_E)=0$. Because $\dot x_0$ is constant, the cross term in the <Euclidean action> vanishes exactly:
$$
m\int_0^{T_E}\dot x_0\dot y\,d\tau
=m\dot x_0[y(T_E)-y(0)]=0.
$$
Consequently
$$
S_{\mathrm{Euc}}[x_0+y]
=\frac{m(x_f-x_i)^2}{2T_E}+\frac m2\int_0^{T_E}\dot y^2d\tau.
$$
Translation of each intermediate integration variable has <Jacobian determinant> one in the <time-sliced configuration-space path integral>. This is an exact <classical-path factorization of a quadratic path integral>, giving
$$
\boxed{\langle x_f|e^{-\beta H}|x_i\rangle
=\mathcal N\exp\!\left[-\frac{m(x_f-x_i)^2}{2\beta\hbar^2}\right],
\quad
\mathcal N=\int_{y(0)=0}^{y(\beta\hbar)=0}\mathcal D y\,
\exp\!\left[-\frac{m}{2\hbar}\int_0^{\beta\hbar}\dot y^2d\tau\right].}
$$
The fluctuation <Euclidean action>, the homogeneous <boundary conditions>, and the measure normalization contain neither $x_i$ nor $x_f$. Hence \b[$\mathcal N$ is independent of both endpoint <positions>]. It depends on $\beta,m,\hbar$.
With the standard position-state normalization, the <Euclidean free-particle bridge amplitude> is
$$
\boxed{\mathcal N=\sqrt{\frac{m}{2\pi\beta\hbar^2}}.}
$$
For example, a <momentum> resolution gives a one-dimensional <Gaussian integral>,
$$
\langle x_f|e^{-\beta H}|x_i\rangle
=\int_{-\infty}^{\infty}\frac{dp}{2\pi\hbar}
\exp\!\left[-\frac{\beta p^2}{2m}+\frac{ip(x_f-x_i)}{\hbar}\right],
$$
whose evaluation gives the same exponent and prefactor. In particular, the result integrates to one over $x_f$ at fixed $x_i$, as required for its <heat kernel> interpretation.
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