Hi,
concerning AAT, I responded in this thread mcu-control-the-antenna-function-of-st25r3916.
Measuring the VDD_RF current is not easily possible on NFC06. You could however measure the power provided to VDD which will include the power going into VDD_RF. I expect the VDD power to be dominated by VDD_RF power once the field is turned on.
As an alternative you could also use the ST25R3916-DISCO board MB1414 where you have the option to bypass the internal regulators and to connect your own power source.
Not all power entering in VDD_RF will exit on the antenna. Depending on matching and antenna you may see also considerable losses on ST25R316 driver and matching components. You would need to either estimate the heat losses or estimate the power inside the carrier field.
Best Regards, Ulysses