Hi, I'm wondering if it's okay for me to use songs that I write for this jam as BGM for a game I am making for another jam, or if that would be against the spirit of the rules?

yeah, as long as the songs have been composed in September you can upload the songs to this jam, and with respect to using them outside this jam, it is also valid as long as you are the one who composed them or you have the permission of who composed them (in this case the first option)