After not finding a solution to the problem anywhere I've worked one out myself. This process will allow you to draw coils in Fusion360 that are not round. On top of that the coil can have any profile you like, not just the few options available in Fusion360.

Step 1: Draw a Cylinder

Draw a Cylinder. This cylinder is only going to be used as a construction body.

You'll want it to be longer than the coil you plan to draw. If it extends above and below your planned coil that will also save you some cleanup at the end.

This cylinder diameter should also fit completely within your planned coil.

Step 2: Draw Your Coil Shape

Next you'll draw a body with the profile of your desired coil. As you'll see I want my coil to be a rectangular tube.

Step 3: Draw Construction Coil

Now, draw a coil to follow the construction cylinder you drew back in step one.

Select one end of the construction cylinder as the drawing plane and use the diameter of that cylinder as the diameter of this coil.

Here it's important to note some of the settings I used in drawing this coil. I find it easiest to use a Triangular (Internal) Section with the Section Position on the Outside.

Since I knew the pitch I wanted my coil to have in the end I set that pitch here.
Note for the future: We are creating the final coil by cutting with this construction coil. That means the resultant coil is going to be the opposite of this construction coil.

Draw this coil as a New Body.

Step 4: Draw "Negative Section" of the Final Coil

Here's the tricky bit that is going to take some planning and forethought.

Use one of the end surfaces of your construction coil as the drawing plane and draw yourself a new shape. This shape will be what gets cut out of the coil profile body you drew earlier.

In my case I want to cut a simple straight path out of my rectangular tube.

Make sure the width of your new shape is longer than the farthest distance of your coil profile body from your construction cylinder.

Step 5: Sweep the Negative Section

Use the Sweep tool to draw a Path + Guide Surface sweep.

Select the negative section you drew in the last step as the Profile.

For the Path of this sweep, select the point of the triangular construction coil that touches the construction cylinder you drew in Step 1.

Select the surface of the construction cylinder as the Guide Surface. This keeps the sweep from twisting in ways you don't want and are hard to predict.

Use a Cut operation.