Pipelining a recursive equation

You will need to make a distinction between pipeline register and filter delay unit(= a register). A filter's delay element is fundamental to its operation but can serve as pipeline by itself. You cannot add any further registers unless very carefully designed.

Your filter has already some delay elements which can serve as pipeline.

Adding any more registers will certainly lead to filter malfunction unless you re-evaluate the functionality.

on the other hand, FIR filters can be pipelined readily in an adder tree but not in the cascaded adders(semi systolic structure). In fact, the systolic structure is an extension of semi that allows pipelining. Similarly, the transposed structure is favoured because the delay elements serve as efficient pipeline.