How to program via ICSP when chip is in-circuit with interfering chips?
Add a suitable resistor between any external circuit that drives an ICSP pin and the AT chip. The resistor must be high enough that the ISP circuit can override the the external circuit, yet low enough that the external circuit can still drive the AT fast enough. You could start with 1k.

An ICSP capability is a combined property of the target chip, the programmer, and the target circuit.