How do I find the proper transfer function of this passive filter?
First of all: use reference designators, it makes addressing components way easier!







simulate this circuit &ndash; Schematic created using CircuitLab

As Chu comments, you cannot separate the circuits as you did with the red boxes. E.g. the impedance right of R2 is not only determined by C2, but by C2||(R3+C3).  

How to solve the circuit?
Formal KVL/KCL approach would work, using voltage divider and/or parallel impedances may also work.
Regarding the latter: you can combine starting from the right side, combining impedences Za = (R3+C3)||C2, working your way back to the left (so, next is Zb = (Za+R2)||C1||R1) and then determine the voltage across R1 and C1 using Iin.
Next, you can use voltage dividing working your way from left to right again (first using Zb and Za, then expanding their terms).

When would it allowed to split the transfer function into blocks?
If the circuit were as follows (with ideal unity gain amplifiers) it would be allowed to split the transfer function in blocks.







simulate this circuit