You are definitely thinking along the right lines when you say combine 3 round-robins. So start with a standard round-robin schedule for 6 players and 5 rounds:
(3 6) (1 2) (4 5)
(5 6) (2 4) (1 3)
(1 4) (3 5) (2 6)
(2 3) (4 6) (1 5)
(2 5) (1 6) (3 4)
Use this for the first 5 rounds, then for rounds 6-10 rotate the 3 columns around 1 position, and rotate again for rounds 11-15.
Round CH LB BP
1 (3 6) (1 2) (4 5)
2 (5 6) (2 4) (1 3)
3 (1 4) (3 5) (2 6)
4 (2 3) (4 6) (1 5)
5 (2 5) (1 6) (3 4)
6 (1 2) (4 5) (3 6)
7 (2 4) (1 3) (5 6)
8 (3 5) (2 6) (1 4)
9 (4 6) (1 5) (2 3)
10 (1 6) (3 4) (2 5)
11 (4 5) (3 6) (1 2)
12 (1 3) (5 6) (2 4)
13 (2 6) (1 4) (3 5)
14 (1 5) (2 3) (4 6)
15 (3 4) (2 5) (1 6)