RSS by custom field

Try this (assuming you have permalinks turned on):
https://www.mywebsite.com/feed/?post_type=game

Doesn’t filter the feed results to the custom field. I results in the same feed items as https://www.mywebsite.com/feed/

I use WordPress multisite. I did some further research and I’m suspecting that it doesn’t work on multisite.

Does somebody know if it should/shouldn’t work on multisite?

I apologize. I misread your original post. I’m not a multisite expert, but I can’t see why this would be possible within multisite.

To start I’d look into creating a custom rss feed inside your theme similar to this:
https://yoast.com/custom-rss-feeds-wordpress/

Then add in your own php to validate the existence of the custom field.

Good Luck.