You can try setting it on registry so your custom profile folder will be used even if Opera isn't started with the customised short-cut.
In this case, I think you'd have to edit the commands in these keys:
HKEY_CURRENT_USER\SOFTWARE\Classes\http\shell\open\command HKEY_CURRENT_USER\SOFTWARE\Classes\https\shell\open\commandMaybe the HKEY_LOCAL_MACHINE ones too/instead or:
HKEY_CLASSES_ROOT\http\shell\open\command HKEY_CLASSES_ROOT\https\shell\open\commandYour changes will be overwritten though the default browser for those is ever modified.
You might be able to edit:
HKEY_CLASSES_ROOT\OperaDeveloper\shell\open\commandfor example instead, so that when Opera is set as the default browser again, all the commands in the other keys are updated too. For stable Opera, that might be:
HKEY_CLASSES_ROOT\Opera.Protocol\shell\open\commandThere are others besides Opera.Protocol like Opera.HTML for .html files. Always look at the command though first. Some of the keys might be for old Opera 12.
Basically command-line options don't help much except for shortcuts unless you change them in the registry too. If you can specify the user data path in a file that opera.exe can read (maybe look at chrome configuration docs), that'd be better.