It kinda turns into a big loop. City doesn't want to put a lot of resources into the disc golf course because it is not used by many people. This is understandable. Disc golfers don't want to play the course because it is never maintained. Also understandable. If there isn't a group available to volunteer to mow/trim it, and the city won't do it, I don't really have any ideas for you.
Hopefully you can get the city to mow it down once as the grass quits growing so you'll have a few months of use.