Hello Damian,
Your initial problem (response with 00) may come from incorrect low power mode entering.
The pull-up resistor on LDP seems to be too high. It is recommended not to exceed 5K. I even recommend to connect it directly to VCC, without pull-up resistor.
Internally, the LPD is pulled-down by a resistor that is automatically disabled when LPD is connected to VCC, in order not to sink current. In order to counter this pull-down resistor (~10k), the pull-up impedance on LPD pin shall be much lower than this pull-down resistor.
I recommend you to use a 2k2 (or max a 4k7) pull-up resistor. As the internal pull-down will be automatically disconnected, the power consumption in low power mode will not increase compared to using a 47k resistor.
With a 47k resistor, the device is probably never entering in low power mode. You can check it easily from RF by reading the EH_CTRL_Dyn register (Read Dynamic Configuration command at address 02h). The bit 3 (VCC_ON) is set to 1 when the tag is powered from VCC and LPD pin is low. When LPD pin is high and the device is in low power mode, this VCC_ON bit is set to 0, even if VCC is present.
You can check with the 47k resistor if this bit remains at 1 when VCC is present.
Best regards.