Firejail manpage says:
--private
Mount new /root and /home/user directories in temporary
filesystems. All modifications are discarded when the sandbox
is closed.
Example:
$ firejail --private firefox
--private=directory
Use directory as user home.
Example:
$ firejail --private=/home/netblue/firefox-home firefox
So this means that Firefox will see /home/user/Desktop as /home/user/, and you should be able to store downloaded files in /home/user/Desktop without doing anything else (assuming default profile).
I have tested it with firejail 0.9.58.2 with the default profile plus the --private=/home/user/Desktop flag on Debian 10.
If Firefox is automatically saving it to the newly created Downloads folder when run with the --private=dir flag, then you should just disable the auto saving feature of downloaded files in Firefox: Preferences -> General -> Files and Applications -> Always ask where to save files. When you do that, and when firefox asks for where to download, the folder that you see as user (home dir) in firefox is actually /home/user/Desktop.
Edit: It is normal that the Firefox settings get reset every time you start in --private=dir, because to Firefox this is your home folder and it looks for the config files .mozilla. If it cannot find it, it will create one for you, with the default settings. Invoke firefox normally with /usr/bin/firefox, then change the download setting I mentioned. After that copy ~/.mozilla directory to /home/user/Desktop and finally you can run firefox with the --private=/home/user/Desktop option.