I think you are trying too hard. On the event {Z_{n-1}=N} the sum is from 1 to N. On the event {Z_1=C_1,...,Z_{n-1}=N}, the sum is again from one to N.

It might help to prove this using the c.f. :- prove that the process can be expressed as a function of Zn and an random variable U(n+1) independent with Zn <--- the problem here is that either Zn is in the index of the independent rv, either I have an infinite number of random variables. So I don't really know how to extend it...

By the tower law,

Characteristic functions uniquely determine distributions and the above is a function of phi and Z_n.