Why don't have a little tournament, of few matches -all rounds in neutral stadium- where to involve the 14 league teams?
So basically, the top 2 would play directly in QFs so the draws can look something like:
1-to QF
8vs9th
3vs14th
6vs11th
2-to QF
7vs10th
4vs13th
5vs12th
4 Matches using the table we have after the 13th Round of league.