Think this could work - for example with the following constraints.
1 Clan in one tournament at a time
Max 8 clans per tournie = 3 rounds usually, occasionally a fourth if tie means both clans progress as per current tournie rules.
2 games each player, 4 players from each clan - tournament to be 'banded' by average rating:
e.g. 1500 - 1700 average clan rating. So long as the 4 players fall into that band, then the clan can be entered (given earlier constraint).
Would make interesting choices e.g. two 1900's and two 1400s against 4 1700's?
Lower bandings could be set for weaker clans. e.g. Max 1200 etc.
Nice to have a tournament sword icon to go with clan league shield on clan setting page for each user. Edit - so members can opt in / out of clan tournies.
I bet this has been discussed before at length, but why not consider again?
Edit also - I like this as it'd be more suited to less active clans like mine, those that don't want to enter teams into the leagues and therefore get swamped with games, or can't due to smaller membership.