This method increasingly resembles a breakthrough. To see why, we could try to quantify the progress made.
The previous “classical” method was to compare \delta(v) for v \in V with
{\rm MMB}(V) = \max_{v \in V} \min_{w \in rot(v)} \beta(w)
instead of
{\rm MMOBD}(V) = \max_{v \in V} \min_{w, u \in rot(v)} \text{odd}(|\beta(w)-\beta(u)|).
The minimal \beta(w) in {\rm MMB}(V) is reached when starting from the smallest term of the cycle.
If we consider the set V of all vectors of length l and weight s for a fixed pair (l,s) such that s \simeq (\log_3 2)l, it is expected that
{\rm MMB}(V) \simeq s 3^{s-1},
the worst case being the high cycle.
According to your figures, {\rm MMOBD}(V) lies somewhere between 3^s and 3^s/l, which is also what is expected for \delta when assuming an optimal lower bound at Roth’s level as recently discussed in this thread. It is getting really tight. And this could be sufficient to rule out cycles for an infinite set of lengths l.
But turning this numerical study into a rigorous proof will not be easy. So I like your proposition of settling the case of smaller collections V first.