As already indicated by Bjorn, this is hopeless. Whether or not a solution $\mu$ to a moment problem has an absolutely continuous part is decided exclusively at infinity; it does not depend on finitely many moments.

So you can always have two measures, one absolutely continuous and the other one purely singular, that have the same moments on an arbitrarily long initial piece. It is perhaps best to think of this in terms of the recurrence
$$
a_n p_{n+1}(z) + a_{n-1} p_{n-1}(z) + b_n p_n(z) = zp_n(z)
$$
satisfied by the polynomials orthogonal with respect to your measure. Knowing the first $N$ moments is the same as knowing the first $N$ coefficients $a_n,b_n$. (This also shows that it's easy to make the supports subsets of $[0,1]$, by controlling the $a$'s and $b$'s.)

So $\alpha_p=\infty$ for any $p>1$ and $\alpha_1=2s_0$. (Even if you insist that both measures are absolutely continuous, the same conclusions hold by approximation.)

By the way, the collection of measures with (the first) $N$ prescribed moments can be described. Search for *Nevanlinna parametrization.*