= Solution
Translation invariance supplies the momentum-conserving delta function. Lorentz invariance, symmetry of $T_{ij}$, stress-tensor conservation, and tracelessness then fix the remaining tensor structure to the transverse traceless spin-two projector shown in the question, up to an overall theory-dependent coefficient. Since the <stress-energy tensor> has scaling dimension $d$, its momentum transform has dimension zero. The delta function has momentum dimension $-d$, so scale invariance requires
$$
\boxed{\beta=d.}
$$
For the Wightman function, the spectral condition restricts support to the appropriate future-directed timelike momenta, with convention-dependent distributions on the null boundary. It vanishes for spacelike momentum. A time-ordered or Euclidean correlator is obtained by analytic continuation and exists more broadly, but polynomial contact terms are renormalization-scheme dependent; in even dimensions the nonlocal power is accompanied by a logarithm.
In a large-$N$ holographic CFT, the single-trace state $T_{ij}(p)|0\rangle$ is dual at leading order to a one-graviton bulk state with the matching boundary momentum and polarization. Multiparticle intermediate states and graviton interactions enter at subleading orders in $1/N$.
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