Sounds like a fun shoot.
If you can get someone to help, I'd use a reflector to throw some extra light on the underside of your subjects. The downward skylights will be perfect for casting light that can be reflected.
With indoor shots it's always the overhead lights and skylights that cause problems. You have mixed light temperature - which can be a problem but is complicated. Just use Auto White Balance.
The second issue - and more of a problem - is blown out elements in your images. This creates two problems - it throws off the exposure and creates blown out areas. You'll need to compensate with those scenes. I'd probably meter with aperture priority then switch to manual exposure since the light on your subject will not be changing very much. Frame your composition to avoid the blown out skylights or light fixtures - or make the blown out area a creative element.
My first choice for composition would be to use a longer lens to isolate your subject without skylights in the image. But don't hesitate to experiment. Use the arms,legs, and shoulders as leading lines to emphasize your subjects. Focus on the eyes or a specific subject element.
Crank up your ISO as high as needed. You should have no problem at ISO 800 and even ISO 1600 is better than soft images.
Get a cooperative subject and make them part of the shot. They can pose or perform a move multiple times to allow you to get the shot.
The problem you had with flash is like photographing snow - the light is extra bright on close particles. I don't see any problem with using an off camera flash. Instead of a flash, you can use a really bright shop light - just be aware of the potential impact on color temperature. You can also bounce the light to spread it or soften it. Bouncing white light off a gold reflector can create a great warm lighting effect.