Bounty update: this can be solved by change of basis, but I'm intrigued by David's solution relying on Fourier Transform of Dirac Delta function, so the bounty is for whoever finds a way to fix his solution to give the right result.
Suppose I have a non-negative real-valued function over $d$-dimensional real vectors as follows
$$f(\mathbf{x})=\exp(-\mathbf{x}' A \mathbf{x})$$
Where $A$ is some symmetric positive definite $d\times d$ matrix. What is the normalization factor to turn this into a valid density over the following set?
$$S_d=\{(x_1,\ldots,x_d)\in \mathbf{R}^d | \sum_i x_i=0 \}$$
Below, David Bar Moshe gives a general solution to computing that integral over space orthogonal to some vector $v$, but I suspect it has a mistake because the answer depends on the norm of $v$.
In particular, suppose $A$ is $d$-by-$d$ identity matrix. Let $v$ be a vector of all ones. Because of symmetry, integrating over space orthogonal to $v$ should be the same as as $d-1$ dimensional Gaussian integral, ie $\pi^{(d-1)/2}$, whereas David's solution gives
$$\frac{\pi^{(d-1)/2}}{d}$$