Do you remember if you tried the following?  Though it doesn't happen,
suppose anyways that P_n is divisible by p in E(K(P_n)), for all n.
Then you can divide all of the Heegner points by p, and get a new
collection {(1/p)*P_n : n in SomeSet}, that might again give rise to
an Euler system of cohomology classes.  If again the (1/p)*P_n are all
divisible by p, divide everything again by p.  You can only do this
finitely many times, because E(K(P_n)) is a finitely generated abelian
group.   Now apply your theorem.