Next game I am planning to make a theme party. I will play a party composed of the 3 bard kits.

The problem is how to distribute the armor.

Which are the armors in SoA a bard can use while still being able to cast spells? I only know of these:
1) Melodic Chains in the Planar Prison
2) Bracers of AC3
3) Bracers of AC4 (anything above is not worth mentioning)
4) Drow Elf Chain Mail in Underdark (but gets disintegrated afterwards)

I also have heard about some elven chain the chief of the shadow thieves is using. Also in Trademeet I can find an other set. But is it possible to cast spells with them?

It's possible to cast spells using any elven chain mail. There is the melodic chain found in the planar prision, and bladesinger chain found in suldenesselar and you can get some non enchanted elven chain from trademeet if you take the evil path for the duelling families quest.

Spoiler
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As you are exiting the Underdark in Chapter 5, you will see Drow fighting Elves. Do not help the Elves, and let the Drow kill them, then kill the Drow and loot as many Elven Chainmail as you can/need. You will not lose reputation points so long as you do not hurt the Elves yourself.

You could also go to the surface and wait around.. Drow swarm from the caverns from time to time, and they're all equipped with elven chain (non-adamantine).

Someone should invent a coating that saves adamantine items from destruction. At least you should be able to wear it at night and store it in the bag of holding in daylight (at least until you use the daystar sword ...)