Yes, you can run with persistence to achieve this. I recommend Ventoy highly for this. It allows a bit more freedom in Persistent Distro Use.
This sounds like information that is stored in the EFI partition, to me. This may be trickier, since Zorin OS will show up as Ubuntu in this field.
An option may be to simply re-install Zorin OS as normal after removing all Ubuntu entries.
Boot LiveUSB, then open a terminal and run efibootmgr
This will list all boot entries in the EFI partition. It should look similar to mine:
BootCurrent: 0000
Timeout: 1 seconds
BootOrder: 0000,0004,0006
Boot0000* ubuntu
Boot0004* MATSHITA DVD-RAM SW830
Boot0004_BbsIndex*
Boot0006 ST2000DM008-2FR102
Boot0006_BbsIndex
Bootffff_BbsIndex*
In my case, if I wanted to remove the Zorin Boot entry, I would focus on Boot0000. You only need the last number of the boot identifier...
So in terminal, I would run this to remove that entry:
efibootmgr -b 0 -B
Let's say that it showed Ubuntu as Boot0003:
efibootmgr -b 3 -B
This can be done by editing grub and adding the parameter fsck.mode=skip just before "quiet splash":
GRUB_CMDLINE_LINUX_DEFAULT="fsck.mode=skip quiet splash"