I would like to get expressions for the time-dependent electric and magnetic fields in the region around two moving point particles.
I can't see how this is possible. Clearly you can't use Gauss' Law or the Biot-Savart Law because the situation is not static.
You can't just sum the two field solutions for particles in vacuum, as this describes a non-interacting system.
If you try to use the retarded potentials then you need to know the past history of the particles, which in turn depends on the electromagnetic fields, which is the thing you're trying to calculate! You might say if the particles are sufficiently far apart, then they travel in straight lines...but their trajectories would only be straight in the limit that they are an infinite distance apart, in which case they would take an infinite amount of time to begin to interact. Perhaps you can somehow integrate from minus infinity in time up to the moment you're interested in.