Reference for exponential Vandermonde determinant identity

Write $\beta = \lambda_n$, the top row of your GT patterns. It's a theorem of [Baryshnikov] that if we choose a uniformly random point in the polytope GT${}_\lambda$, it's equivalent to choosing a Haar-random Hermitian matrix with spectrum $\beta$ and then taking its "principal minors". (I've also seen this fact credited to Weyl, and others.) More precisely, let $B = \mathrm{diag}(\beta)$, and form a matrix $X = U B U^\dagger$, where $U$ is a random unitary. Then let $\lambda_{11}$ be the top-left entry of $X$, let $\lambda_{21}, \lambda_{22}$ be the eigenvalues of the top-left $2 \times 2$ submatrix of $X$, ..., and let $\lambda_{n1}, \dots, \lambda_{nn}$ be the eigenvalues of the top-left $n \times n $ submatrix of $X$ (namely, $\beta$). Then $\lambda$ is uniformly random in the polytope GT${}_\lambda$.

This probability distribution on $\lambda$ is basically your integral, but we have to divide by the volume of the polytope, which is $V(\lambda)/[(n-1)! (n-2)! \cdots 2! 1!]$. I guess this is standard? If not, it's also in Baryshnikov.

Having done so, your identity is the Harish-Chandra--Itzykson--Zuber identity applied to the matrices $A = \mathrm{diag}(\alpha)$ and $B$. This follows by inferring the diagonal entries of $X$ from the Gelfand-Tsetlin pattern $\lambda$, which you can do because the Gelfand--Tsetlin pattern gives you the traces of all the top-left submatrices.

(By the way, I think the [Faraut] paper referenced below has a good exposition of some related things.)

Baryshnikov, Yu., GUEs and queues, Probab. Theory Relat. Fields 119, No.2, 256-274 (2001). ZBL0980.60042.

Faraut, Jacques, Rayleigh theorem, projection of orbital measures and spline functions, Adv. Pure Appl. Math. 6, No. 4, 261-283 (2015). ZBL1326.15058.

This looks like a special case of a formula by Samson Shatashvili related to the HCIZ integral as mentioned in Ryan's answer. Compare, in particular the two ways of computing $\langle 1\rangle$ given by Equations 3.2 and 3.4 in "Correlation Functions in The Itzykson-Zuber Model" (thanks to Leonid Petrov for letting me know about this reference in his answer to this MO question).