Programming an atmega328p via raspberry pi gpio and avrdude incompatible with logic level shifter?

I don't see a problem, so I assume that a signal is too weak or too slow with the level shifters.

To make the level shifters work, all the LV must get 3.3V and all the HV must get 5V. That seems okay.
Is the signal for the reset also going through the level shifter ? That might be a problem.
You could add a few decoupling capacitors for the 3.3V and 5V power on each breadboard.
The GND path is going through a number of wires and breadboard connections. You could connect all GND to a single strip on the breadboard.