While it would take a substantially more work than creating a persistent USB you would have to create a LiveCDCustomized Media. The link is for a starting point so that you can see some of what is involved.
The default LiveUSB is designed to allow the user to easily Try Ubuntu, but it's main design is for the installation, and of course, having a variety of drivers for different computer's hardware.
You could spend some time in the environment of the Live Session, mounting the space you see and moving files around to the other space, as well as other drives of your computer. But that will take a substantially lot of work also. Some of the things you might want to configure may require rebooting. That would mean that when you reboot, you will have to start from scratch.
By design its not setup for the type of task you describe. The space that you see available may eventually be used for the intended task as the Operating System and it's necessities grow with time.
You can use the other space, but you would have to tedious manually map the space for usage and move files from root /
to the mapped space. Then linked to them for access.
The unused/unmaped space isn't available by default and design. You run out of space because the actual root /
is in RAM, not the actual USB.
The purpose of the persistence that you don't want to use is to allow the root /
system to be on the actual disk where the space is located.