Including the custom installed Chromium as an option in the "Default Applications" dialog
The list of default applications in the "Default Applications" dialog is generated based on the information about mime types the application can handle. That information is provided through the
.desktop launcher of the application.
Thus, create a valid
.desktop launcher for your application. Include a line
MimeType= to provide the mimetypes it can handle and include at least
x-scheme-handler/ftp;x-scheme-handler/http;x-scheme-handler/https; for it to become included in the dialog.
Creating a symlink to the custom installed Chromium
This is simply a matter of creating a standard symbolic link in a directory that is included in your search path. Conventionally, a good location is
/usr/local/bin where executables are kept that are manually installed, outside of the regular package management systems (currently the traditional APT and the newer snap systems in Ubuntu).
sudo ln -s /path/to/executable /usr/local/bin/chromium-browser
would create a symbolic link
chromium-browser, allowing you to run the executable
/path/to/executable by typing "chromium-browser" in the terminal.