Cobalt chloride decomposes only at extremely high temperatures and it melts only at very high temperatures (726 °C), which could not be reached, in any case not before all water in the silica gel would be converted to steam and it would be eliminated. Even when no water is left, it is unlikely that the beads with cobalt chloride could absorb enough microwave energy to be heated at very high temperatures.

So by itself cobalt chloride could not cause any problem.

However, I have no idea whether the cobalt chloride is not mixed with some organic binder, to make it stick to the silica gel beads, which could burn in the oven, though that is also unlikely to happen before all water is removed from the gel, allowing an increase in temperature above the boiling temperature of water.

By using low microwave power and short time, so that no boiling of the contained water should be seen, it should be possible to dry even beads with cobalt chloride.

Are there studies on whether it sheds dust and under what conditions?

Also, where are people even still getting cobalt chloride gel? Do they still make it? I sure wouldn't buy any.

I wouldn't even buy the orange to green stuff by choice these days for anything DIY, it's still too toxic when mechanical hygrometers are cheap.