Answering my own question; turns out that the index additivity, "reduction to FT" and unitarity conditions are not necessary at all. Ignoring said conditions-
Suppose $T_a$ exists and satisfies all of the other properties. Then, the kernel of $T_a$ can be represented as a tempered distribution $K_a(x,y)$. Let $H$ denote the Hermite-Gaussian function, normalised to "ordinary frequency". Then, $K_a(x,y) = Σ_n,_m H_n(x)H_m(y)Q_n,_m(a)$ for some function $Q_n,_m$ of $a$.
Let $Q_n,_m(a)=exp(iM_n,_m(a))$, $M$ not necessarily being real. Applying the condition that $d/daK_a[f](y) $$= i(1/8d^2/dy^2-π^2y^2/2+π/4)K_a[f](y)$ to the Hermite expansion of both sides and appreciating that $(1/8d^2/dy^2-π^2y^2/2+π/4)$$H_m(y)$ $=-πm/2$$H_m(y)$, $d/daM_n,_m(a)$$=-πm/2$ and $M_n,_m(a)=-πma/2+S_n,_m$, $S$ being constant of $a$.
Now throw in the "reduction to the identity" condition- $δ(x-y)=Σ_m H_m(x)H_m(y)=Σ_n,_m H_n(x)H_m(y)e^{iS_n,_m}$. Since Hermite expansion coefficients are unique, and $S_n,_m$ is constant of $a$, $e^{iS_n,_m}$ must be $δ_{nm}$ for all $n$, $m$.
Therefore, $K_a(x, y) = Σ_m H_m(x)H_m(y)(-i)^{am}$ for all $a$, which is exactly the definition of the FRFT. Q.E.D.