I am not sure if I understand your question correctly. So basically you want to compute the sum of all combinations of Beta and Alpha right? This can go into the generated quantities block, if you don’t want to use your mixture variable somewhere else in the Stan file. How about:
generated quantities {
vector[6] mixture;
mixture[1] = Alpha[1] + Beta[1];
mixture[2] = Alpha[1] + Beta[2];
…
}