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This is not an answer, but a derivative question worth pursuing. I put it here in hopes someone can complete it to an answer which solves the posted question.

Let p be a prime dividing n+1. Divide a given set A of integers into residue classes mod p, so there are a_j many members of A equal to j mod p. When I feel up to it, I will write out the equations for how many of the sums of unordered pairs from A add up to a number which is k mod p. The upshot is that for a set A to satisfy the required conditions mod n(n+1)/2, it needs to satisfy an equidistribution system of equations mod p for each such p. In other words, if S_c is the count of such sums from A with sum equal to c mod p, then S_c=S_b for c different from b mod p. We have p choose 2 equations of a form like $$ \sum_{j+k=c \bmod p, j \lt k} a_ja_k + \sum_{j+j=c \bmod p} (a_j^2 + a_j)/2= \sum_{j+k=b \bmod p, j \lt k} a_ja_k + \sum_{j+j=b \bmod p} (a_j^2 + a_j)/2$$.

The question now is are there any tuples of nonnegative integers $a_j$ whose sum is $n$ and which satisfy the above system? For n=2 and p=3, we have (0,1,1). Are there any others?

It is easy to verify (by running through 3-partitions of 5) that for n=5 and p=3, there are no (error: at least one, thanks David Speyer) 5- sets A which sums are equidistributed mod 3.

Gerhard "Is This In The Literature?" Paseman, 2018.06.19.

This is not an answer, but a derivative question worth pursuing. I put it here in hopes someone can complete it to an answer which solves the posted question.

Let p be a prime dividing n+1. Divide a given set A of integers into residue classes mod p, so there are a_j many members of A equal to j mod p. When I feel up to it, I will write out the equations for how many of the sums of unordered pairs from A add up to a number which is k mod p. The upshot is that for a set A to satisfy the required conditions mod n(n+1)/2, it needs to satisfy an equidistribution system of equations mod p for each such p. In other words, if S_c is the count of such sums from A with sum equal to c mod p, then S_c=S_b for c different from b mod p. We have p choose 2 equations of a form like $$ \sum_{j+k=c \bmod p, j \lt k} a_ja_k + \sum_{j+j=c \bmod p} (a_j^2 + a_j)/2= \sum_{j+k=b \bmod p, j \lt k} a_ja_k + \sum_{j+j=b \bmod p} (a_j^2 + a_j)/2$$.

The question now is are there any tuples of nonnegative integers $a_j$ whose sum is $n$ and which satisfy the above system? For n=2 and p=3, we have (0,1,1). Are there any others?

It is easy to verify (by running through 3-partitions of 5) that for n=5 and p=3, there are no (error: at least one, thanks David Speyer) 5- sets A which sums are equidistributed mod 3.

Edit

It turns out there are others. Let d be an odd divisor of n+1. One can look at the addition table of 0,..,n to see that there are an equal number of representatives mod d among all ordered pairs of sums. As d is odd, this reduces to equidistribution mod d among the unordered pairs of their sums. "Removing" the last column (addition by n) shows that the equidistribution remains when restricted to the set A=(0,1,...,n-1). So there is a solution to the system of equations above, not just for odd primes p, but also for odd divisors d of (n+1) (and for odd d dividing n). So the approach suggested above does not directly lead to a proof of the nonexistence of such sets mod m (because p=d is too small). It might yield something for large divisors d of m, however.

End Edit

Gerhard "Is This In The Literature?" Paseman, 2018.06.19.

This is not an answer, but a derivative question worth pursuing. I put it here in hopes someone can complete it to an answer which solves the posted question.

Let p be a prime dividing n+1. Divide a given set A of integers into residue classes mod p, so there are a_j many members of A equal to j mod p. When I feel up to it, I will write out the equations for how many of the sums of unordered pairs from A add up to a number which is k mod p. The upshot is that for a set A to satisfy the required conditions mod n(n+1)/2, it needs to satisfy an equidistribution system of equations mod p for each such p. In other words, if S_a is the count of such sums from A with sum equal to c mod p, then S_c=S_b for c different from b mod p. We have p choose 2 equations of a form like $$ \sum_{j+k=c \bmod p, j \lt k} a_ja_k + \sum_{j+j=c \bmod p} (a_j^2 + a_j)/2= \sum_{j+k=b \bmod p, j \lt k} a_ja_k + \sum_{j+j=b \bmod p} (a_j^2 + a_j)/2$$.

The question now is are there any tuples of nonnegative integers $a_j$ whose sum is $n$ and which satisfy the above system? For n=2 and p=3, we have (0,1,1). Are there any others?

It is easy to verify (by running through 3-partitions of 5) that for n=5 and p=3, there are no 5- sets A which sums are equidistributed mod 3.

Gerhard "Is This In The Literature?" Paseman, 2018.06.19.

This is not an answer, but a derivative question worth pursuing. I put it here in hopes someone can complete it to an answer which solves the posted question.

Let p be a prime dividing n+1. Divide a given set A of integers into residue classes mod p, so there are a_j many members of A equal to j mod p. When I feel up to it, I will write out the equations for how many of the sums of unordered pairs from A add up to a number which is k mod p. The upshot is that for a set A to satisfy the required conditions mod n(n+1)/2, it needs to satisfy an equidistribution system of equations mod p for each such p. In other words, if S_c is the count of such sums from A with sum equal to c mod p, then S_c=S_b for c different from b mod p. We have p choose 2 equations of a form like $$ \sum_{j+k=c \bmod p, j \lt k} a_ja_k + \sum_{j+j=c \bmod p} (a_j^2 + a_j)/2= \sum_{j+k=b \bmod p, j \lt k} a_ja_k + \sum_{j+j=b \bmod p} (a_j^2 + a_j)/2$$.

The question now is are there any tuples of nonnegative integers $a_j$ whose sum is $n$ and which satisfy the above system? For n=2 and p=3, we have (0,1,1). Are there any others?

It is easy to verify (by running through 3-partitions of 5) that for n=5 and p=3, there are no (error: at least one, thanks David Speyer) 5- sets A which sums are equidistributed mod 3.

Gerhard "Is This In The Literature?" Paseman, 2018.06.19.

This is not an answer, but a derivative question worth pursuing. I put it here in hopes someone can complete it to an answer which solves the posted question.

Let p be a prime dividing n+1. Divide a given set A of integers into residue classes mod p, so there are a_j many members of A equal to j mod p. When I feel up to it, I will write out the equations for how many of the sums of unordered pairs from A add up to a number which is k mod p. The upshot is that for a set A to satisfy the required conditions mod n(n+1)/2, it needs to satisfy an equidistribution system of equations mod p for each such p. In other words, if S_a is the count of such sums from A with sum equal to c mod p, then S_c=S_b for c different from b mod p. We have p choose 2 equations of a form like $$ \sum_{j+k=c \bmod p, j \lt k} a_ja_k + \sum_{j+j=c \bmod p} = \sum_{j+k=b \bmod p, j \lt k} a_ja_k + \sum_{j+j=b \bmod p} $$.

The question now is are there any tuples of nonnegative integers $a_j$ whose sum is $n$ and which satisfy the above system? For n=2 and p=3, we have (0,1,1). Are there any others?

It is easy to verify (by running through 3-partitions of 5) that for n=5 and p=3, there are no 5- sets A which sums are equidistributed mod 3.

Gerhard "Is This In The Literature?" Paseman, 2018.06.19.

This is not an answer, but a derivative question worth pursuing. I put it here in hopes someone can complete it to an answer which solves the posted question.

Let p be a prime dividing n+1. Divide a given set A of integers into residue classes mod p, so there are a_j many members of A equal to j mod p. When I feel up to it, I will write out the equations for how many of the sums of unordered pairs from A add up to a number which is k mod p. The upshot is that for a set A to satisfy the required conditions mod n(n+1)/2, it needs to satisfy an equidistribution system of equations mod p for each such p. In other words, if S_a is the count of such sums from A with sum equal to c mod p, then S_c=S_b for c different from b mod p. We have p choose 2 equations of a form like $$ \sum_{j+k=c \bmod p, j \lt k} a_ja_k + \sum_{j+j=c \bmod p} (a_j^2 + a_j)/2= \sum_{j+k=b \bmod p, j \lt k} a_ja_k + \sum_{j+j=b \bmod p} (a_j^2 + a_j)/2$$.

The question now is are there any tuples of nonnegative integers $a_j$ whose sum is $n$ and which satisfy the above system? For n=2 and p=3, we have (0,1,1). Are there any others?

It is easy to verify (by running through 3-partitions of 5) that for n=5 and p=3, there are no 5- sets A which sums are equidistributed mod 3.

Gerhard "Is This In The Literature?" Paseman, 2018.06.19.