To upload a new firmware image on the RP2350 (the "pico" part of "pico2-ice"), you need to switch to that RP2350 UF2 bootloader mode instead of the FPGA UF2 mode.

You can do that by connecting a jumper between the BT pin a Gnd pin or hold the SW1 button, then toggle power:

Connect the BT pin to ground or hold the SW1 button. For instance with a pair of tweezers, a jumper, or an uncoated paper clip.
Toggle power to the RP2350, for example by unplugging and re-plugging the USB cable.
Disconnect the BT pin from ground.
Look for an USB drive named RP2350 and mount it, open it, and check that there are only 2 files in it: INFO.HTML and INFO_UF2.TXT.
Copy the .uf2 firmware file onto the USB drive.
Eject the USB drive. At this step, the firmware should be uploaded and running.

photo of button near the shortable pins

Since version v1.6.1, it is possible to reboot the pico2-ice into bootloader "BOOTSEL" mode without jumpers:

picocom --baud 1200 /dev/ttyACM0

This uses the convention established by Arduino boards which reset into bootloader mode when connecting with baud 1200.

After this, it becomes possible to use the picotool to control the firmware (upload, check version, reboot):

$ sudo picotool info
Program Information
 name:      pico_ice_default
 features:  USB stdin / stdout

Troubleshooting

RPI-RP2 is still visible after step 6

This means the device remained in "bootloader" mode. One thing to check is that the BT pin is not connected to anything anymore, then try to reset the board.