Relativity of simultaneity - An example
Your logic is correct but for one misunderstanding right at the beginning. The observer is smart enough to know that just because the photons are received simultaneously doesn't mean they were emitted at the same time. The emitting of a photon and the receiving at a different time and different place are two distinct events, as explored further in this post, among others.
Basically, you need to find a position and a velocity such that the observer infers that the photons were emitted simultaneously. The observer makes this inference by saying the travel time for a photon is the distance between his current position and the source, as measured in his moving frame, divided by $c$.
In fact, you don't even need to solve for the position to get the velocity, but I'll let you crank through the appropriate Lorentz transformations.