Unfortunately there is no such schedule. The argument goes as follows:
There are 55 matches to play in the round robin, 27 of these are played on day 1, the remaining 28 are played on day 2. Consider the 28 matches on day 2, which involve 28*2=56 slots in which a team plays. Within the rules, 10 of the 11 teams can each be assigned 5 slots each, but the remaining team must play in all 6 remaining slots if the round robin is to be completed in 2 days.
Here is an example of such a schedule:
M1 M2 M3 M4
(I J) (B D) (E F) (---)
(B H) (D G) (C F) (A K)
(B E) (A I) (J K) (C D)
(A C) (F J) (G H) (B I)
(A G) (F H) (E K) (D J)
(C I) (E J) (B G) (A H)
(F K) (C G) (E I) (D H)
------------------------
(B K) (C H) (A D) (E G)
(D E) (H I) (A J) (B F)
(E H) (G K) (C J) (F I)
(H J) (D K) (A B) (C E)
(F G) (B J) (D I) (C K)
(D F) (A E) (H K) (G I)
(I K) (A F) (B C) (G J)
teams A to J play exactly 5 times each on both days, but team K plays 4 times on day 1 and 6 times on day 2. Additionally each teams plays 2 or 3 times on each mat.
Hope that helps.
That's OK, I thought there might be no flexibility in the school rules, so here is a 10 team version.
M1 M2 M3 M4
(B I) (F G) (D H) (E J)
(A D) (B F) (C G) (---)
(A F) (D J) (C E) (H I)
(G J) (A H) (E I) (B D)
(A E) (---) (F J) (B C)
(B H) (D G) (I J) (A C)
------------------------
(G I) (C J) (A B) (E F)
(D I) (E H) (---) (B J)
(C D) (B E) (F H) (A G)
(E G) (C H) (A J) (F I)
(H J) (C I) (B G) (D F)
(C F) (A I) (D E) (G H)
There need to be 3 rounds where one of the mats is not used, so I have put these on different mats and at least one on each day. Teams should wrestle 2 or 3 times on each mat, and any one team will have one bye on one of the days, and two byes on the other day. Where a team has two byes, then they will not be consecutive. Good luck with the tournament.