Generating numbers to be averaged efficiently
A reasonable solution is to first generate a random point in the simplex
$$ x_1 + \cdots + x_K = 1, \qquad x_1,\ldots,x_n \geq 0. $$
Check out this question for several ways to do this. Then take
$$
P_i = \frac{P}{\alpha_i} x_i.
$$
This should give you a uniformly random point in the geometric body you are interested in
$$
\sum_{i=1}^K \alpha_i P_i = P.
$$