Thanks for getting back to me, Dries.
Our tag is not battery powered. It is encapsulated in a material, which makes it physically inaccessible. In the actual use case it can only be interacted with via the field.
I am using a Moto Z 2 Force, and my collaborator is using a Google Pixel 2. We have much experience with positioning the tag antenna to couple with the phone antennas, so I'm confident we have optimal coupling. I do have an ACR122U which can generate a more powerful field, but I'm not sure how to load the firmware using that device.
Using the NHS3100TEMODB demo PCB, I can load the firmware images correctly using SWD. I do not know a way to put the demo PCB back into "One-time NFC program downloader" mode. Is the original firmware image the ICs ship with publicly available? If that were the case I could test this using the demo PCB, which has a better coil.