Will 300W lights be too much for a 200W timer box?
Is the controller in jeapordy of having any dangerous meltdowns or will the lights just run a little dimmer?


Yes. By yes I mean without knowing exactly how the controller is built it is hard to say. It is likely a mechanical relay or a solid state relay switching the lights on and off. In that case, there is no limiting the power drawn, and the higher wattage can lead to premature failure of the controller's relay. Mechanical relays will weld and the relay contacts will fail. Solid State Relays will release the magic smoke. It is unlikely to catastrophically fail (i.e. house fire) but it is possible.

If its rated for less than your load, you already know it won't end well. Best case, the controller fails before it's standard life time. Worst, you will be outside while the fire department is hosing down your house. Or worse.