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.