At the moment copyto! with Bidiagonal is O(n^2):
julia> n = 1000; B = Bidiagonal(randn(n), randn(n-1), :L); @time copyto!(B, B);
0.002047 seconds (3 allocations: 16.031 KiB)
julia> n = 10_000; B = Bidiagonal(randn(n), randn(n-1), :L); @time copyto!(B, B);
0.127239 seconds (5 allocations: 156.375 KiB)
julia> n = 100_000; B = Bidiagonal(randn(n), randn(n-1), :L); @time copyto!(B, B);
11.586487 seconds (5 allocations: 1.526 MiB)