There’s no method in WordPress’ UI for adding CSS for a single page only, if that’s what you’re asking for. There might be plugins available that offer it, though, but discussing that would be off topic here (I’d suggest a local community group or similar if you’re asking for plugin recommendations).
The simplest “proper” way to do this would be using the body class to target the specific page, as you mentioned. It’s not a flaky hack. If you’re truly cloning the site between environments then the IDs would be the same so there shouldn’t be any issue.
The more robust solution, which I would expect if you have a development workflow that involves multiple environments, would be to have a proper child theme and you would make whatever template changes needed to add a new class to that element on that page and then add the appropriate CSS to the theme.