I think this is a bit different likelihood. A mixture of binomials describes a process of selecting a random coin from a bank and tossing it N times, not selecting N coins and tossing them once.
If I’m not overthinking, this might do the job:
p_A = wA*p1 + (1-wA)*p2
p_B = wB*p1 + (1-wB)*p2