It is indeed impossible as stated. But a different tweak to the format of play may also be workable. Play the first 4 rounds as follows:
round venue1 venue2
1 (6 3 2 4) (8 5 7 1)
2 (7 4 6 5) (2 1 8 3)
3 (1 2 5 6) (3 8 4 7)
4 (5 6 3 8) (4 7 1 2)
If the four teams who meet at a venue are (A B C D), then three sub rounds can be played as follows
sr1 (A v B) (C v D)
sr2 (A v C) (B v D)
sr3 (A v D) (B v C)
For rounds 1 to 3 play all three sub-rounds, however at round 4 omit the sub-round sr1. So by the end of round 4 all teams will have played exactly 11 matches. The remaining 3 matches of the double round robin can be played at a single venue in round 5 and can be resolved into 3 sub rounds as follows:
(1 v 3) (2 v 5) (4 v 8) (6 v 7)
(1 v 4) (2 v 7) (3 v 5) (6 v 8)
(1 v 6) (2 v 8) (3 v 7) (4 v 5)
Hope that helps.