Usually I recommend using mkusb in order to create a persistent live USB drive. But in this case I think it is too risky, because there is no backup, the only working operating system is in the target drive.
Instead I suggest to make the default /var/log
and /var/crash
mounted writable
partition into a 'free' usbdata
partition, that can be mounted and unmounted by the user.
When booting, at the grub menu, press 'e' to make a temporary edit: add nopersistent
to the line starting with linux
linux ... quiet splash nopersistent ---
continue booting with F10 or ctrl x
now you can unmount the partition with ext4 file system
sudo umount /dev/sdx3
where x is the drive letter (can be a, b, c ...)
change the label
sudo tune2fs -L usbdata /dev/sdx3
shutdown, wait for 10 seconds and boot again.
The system will be live (not persistent live), but the ext4 partition can be used for storage. Until you have another drive available this is a rather safe method to create storage space on the drive. In order to make it convenient, you can create one or more directories at its top level and modify the ownership.
The partition will probably automount (it did when i tested with Lubuntu 20.04.4 LTS, which is lighter than standard Ubuntu).
cd /media/lubuntu/usbdata
sudo mkdir Docs
sudo chown 999 Docs
999 is the numeric ID of the live user (for standard Ubuntu as well as Lubuntu and the other community flavours). Now you can go there and test writing files.
cd Docs
echo 'Hello World' > hello
cat hello
ls -l hello