Think of the Black Dragon as a flying Lich. He will attack the unit placed down last that is in his range. That's why he changed targets from Dorca to Calliope when she was placed down. The strategy here is to have the black dragon attack Dorca, then a witch placed in the top area, and then the bait units at the west entrance while your other units are killing the other enemy units and dragons. For bait units, use units that have more than 1K health like the bandits and heavy armors. Units with MR will help too. Also, you need to time when to use the archers' skills as the Black Dragon needs to be dropped next to Claudia so she can kill him with her skill activated.
Refer to the wiki page for dragon fang drop rates: http://millenniumwaraigis.wikia.com/..._Dragon_Slayer . All the dragon bosses give you 100% drop rates for their fangs.