I know it's not traditional, but is there any interest or thoughts on multi-clan cards? Magic have cards that require 2 or more types of mana (in addition to generic mana payable with any type).
Such things in this game would be characters who could be played in either clan's dynasty deck (say Isawa Kaede who is Phoenix by birth, but has married Toturi (Lion)). If it were a conflict card it could be splashed along with either clan, but it would be free for either clan.
For example, Mountain Armor a 1-cost +1MIL/+0POL with insert made-up text would be a Dragon/Phoenix attachment. Either clan could add it to their deck without paying influence, but would cost X-Influence for a Lion player who splashed either Dragon or Phoenix.