I'm into cycling and running, and tried to be the guy on the course who does running disc golf rounds and gets in 4-6miles (two quick running DG rounds) of simultaneous exercise and disc golf. I tried it with small north face bladder packs, then realized that it was just easier to run with three/four discs in my hand and map out where the water fountains were.
But I realized that for me, I enjoy the gradual pace of playing in a 2-4 some, watching everyone's discs fly, shooting the sh*tt, taking off and putting back the bag per hole and taking in beautiful scenery + if I'm lucky seeing/throwing beautiful DG flights/putts/upshots.
Lugging around DG specific backpacks/bags is all part of the experience
Its very Zen and communal. Simultaneously stressful and relaxing.
I just run and bike separately and enjoy that as well.