Okay, I think I am satisfied that the "partial answer" I included in the question is the correct answer. The proof of the WW theorem involves matrix elements of the form $$\langle p|T^{\mu\nu}(x)|p'\rangle, $$ where $|p\rangle$ is a momentum eigenstate of the spin-2 particle and $p,p'$ are two nearly equal null momenta. The proof relies on this matrix element being nonzero, while it does equal zero for my case of two collinear photons. The operator $T^{\mu\nu}$ cannot change the momenta of two different photons, because it is only quadratic in the photon field $A^\mu$. 298
Therefore the proof does not apply, and such states can of course exist.
The reason I was dissatisfied with the answer, and called it only "partial", was that I didn't understand why we should be so confident that the above matrix element does not vanish when $|p\rangle$ is a single-particle state such as an "emergent graviton" that is a bound state of other fields. If it can vanish when the underlying particles are not bound, then how do we know that creating a bound state will change things?
Now I think I get it: the condition $\langle p|T^{\mu\nu}(x)|p'\rangle\ne0 $ is basically an intuitive physical requirement. It means that if (hypothetically) there was a an interaction of the form $T^{\mu\nu}h_{\mu\nu}$ (and of course there is such an interaction, namely gravity, but we do not require this), then a background field $h_{\mu\nu}$ with a small gradient could cause slightly changes in the momentum of our particle, making it move along a curved trajectory. This can be taken as a reasonable definition of what is meant by the particle being "charged under $T^{\mu\nu}$", and this is what is required for the WW theorem to apply.
This behavior is also pretty close to what we mean by a state being "bound", namely that the different components maintain a shared trajectory when "pushed around" by mild forces. But it does not describe particles that just happen to be collinear; in that case the force will tend to spread the trajectories slightly apart. 298