Here’s a divisibility-based proof that 11101000 can’t be an integer cycle, regardless of the denominator 
If the cycle members were integers, any sum/difference of them would also be an integer, but
-73+197-383+662+331-584-292+146 = 2^2
and \cfrac{2^2}{\delta} can’t be an integer.
For every de Bruijn sequence with l=8, there exists some signage that results in a 3-smooth sum, indivisible by \delta. (And for l=16, I think.)
But at l=32, there are already exceptions, so this is unfortunately not a general method.