There’s a long history about this topic that is still unresolved to this day.
https://core.trac.ww.wp.xz.cn/ticket/12702
Searching for plugins that maybe do this type of thing for you leads me to https://ww.wp.xz.cn/plugins/sticky-posts-switch/ as at least one example. Worth trying out. Otherwise it’s going to need to be a custom solution to handle.
Thank you. That plugin may not be supported?
“This plugin hasn’t been tested with the latest 3 major releases of WordPress. It may no longer be maintained or supported and may have compatibility issues when used with more recent versions of WordPress.”
Definitely possible that they’re not paying attention to it too much, but it also could just not need much if it’s solid enough. There may also be other plugins that are more current that do something for sticky posts for custom post types as it seems WP Core still won’t do something itself.