Distributing shares in a particular monotone access structure
Well, the easiest way I can see to represent this access structure is:


Select three random values $r_1, r_2, r_3$
Generate a threshold-2 secret sharing scheme with the secret $S$ (where $S$ is the ultimate secret), and whose shares are $\alpha_1, \alpha_2, \alpha_3$
Generate an independent threshold-2 secret sharing scheme with the secret $r_3$, and whose shares are $\beta_1, \beta_2, \beta_3$.


Then,


$A$ gets $\alpha_1$, $r_3 \oplus S$
$B$ gets $r_1$, $\beta_1$
$C$ gets $r_2$, $\beta_2$
$D$ gets $r_1 \oplus r_2 \oplus S$, $\beta_3$
$E$ gets $\alpha_2$, $r_3 \oplus S$
$F$ gets $\alpha_3$, $r_3 \oplus S$


It's easy to see that any of the allowed subsets can recover $S$ (e.g. for the subset $A, B, C$, they jointly know $\beta_1$ and $\beta_2$, that allows them to recover $r_3$; and with $r_3 \oplus S$ gives them $S$).  It should be fairly clear that any subset not within the access structure gets no information about $S$.

This doesn't give an answer to the precise question to raised; it does appear to solve the problem you were trying to address.