In case it's helpful, the code below seems to trigger the same issue.

L = 5;
n = 4;
id = ConstantArray[1, L] // SparseArray // DiagonalMatrix;
D0 = SparseArray[{{i_, j_} /; Mod[j - i, L] == 1 :> 1, {i_, j_} /; Mod[j - i, L] == 0 :> -1}, {L, L}];
Dj[j_] := KroneckerProduct @@ Table[If[i == j, D0, id], {i, n}]
M = Sum[If[a != b != c != d, KroneckerProduct[Dj[b], SparseArray[{a, (c - 1) n + d} -> 1, {n, n^2}]], 0], {a, n}, {b, n}, {c, n}, {d, n}];
svd = M // Normal // N // SingularValueDecomposition;

The original question has been edited to establish that the SVD "cflsvd" bug is sensitive to the least-significant-bits of MachinePrecision input matrices, which is behavior that no well-conditioned SVD algorithm should exhibit.

Please note, too, that an effectively unlimited number of convergence-failing matrices can be generated, as it turns out that randomly permuting rows and columns of a numerical matrix that "cflsvd"-fails, sporadically--not invariably---yields a new matrix that similarly "cflsvd"`-fails.