Do you have to have a filter to keep baby fighting fish alive?

All life (Including baby Bettas) in water create (make) Ammonia. Ammonia is deadly poison and will eventually kill whatever fish life is in the water. A filter uses naturally developed aerobic bacteria to convert this poisonous ammonia first into (still poisonous) Nitrite and then converts that into none poisonous Nitrate. A weekly water change reduces the amount of Nitrate. So the answer is "Yes your baby Bettas should have a cycled filter as should all other fish".