I’m going to start bassoon soon and I was wondering if I could use seat strap for when I’m playing, but a harness when I’m waiting to go on stage and while I’m walking on, so I don’t have to keep hold of it completely. I’d just unhook it when it’s time to play and put it in the seat strap cup.

There's no reason you cannot do that - do what works for you. You might even find using both when sitting is good - some people reckon that way reduces the weight on the left hand & helps to keep the bassoon at a comfortable angle.

I always use both a seat strap and a harness for the very simple reason that it takes nearly all the weight off my left hand and arm.

If you angle the seat strap just right, the bassoon stays in playing position with very little effort. My left hand/arm are somewhat weaker than the right, and before I hit upon this setup, I would leave a rehearsal tired and sore.

Most of my friends use the seat strap straight across the seat of their chair -- parallel to the front edge. I put mine from the back to the front, and the strap "hooks" around the front right corner of the chair "anchoring" the butt joint and counterbalancing the harness.

I like this setup so much that I keep a harness and strap in my case, and another set at my practice corner to minimize the fuss of getting ready to practice.