I don't understand your logic. Or maybe I don't understand your assumptions. To predict the state at T13, you shouldn't start with T1, you merely need to use T12.
From your 1000 x 12 data points, you can build a matrix that gives the empirical probability of transitioning from State k to any other state. (This matrix is formed by aggregating over time, which assumes that the probabilities do not change over time.) So for each customer that you want to predict, you should look at the state that they are in right now (T12). If they are in State k, then the k_th row of the transition matrix is the probability vector for the next State. If you want to predict the next State, your best prediction is the State in row k that has the highest probability.
If you want to predict their state at T14, T15, etc, you can use vector-matrix multiplication to iterate the process. Equivalently, you can look at the rows of P**2, P**3, etc, where P**2 = P*P is the matrix product of the transition matrix with itself, P**3=P*P*P is cubic product, and so forth. At each stage, your best prediction for the next State is the column in the k_th row that has the highest probability.
To score all customers at once, represent their current state in a binary indicator matrix and form the matrix product of the indicator matrix with the transition matrix. Then choose the largest probability for each customer.