Is there a way to examine amount of red/white muscle fibers without biopsy?
Indeed, a paper was published in 2011 regarding the same for humans by using a magnetic resonance spectroscopy to measure carnosine content.

A significant positive correlation was found between muscle carnosine, measured by 1H-MRS, and percentage area occupied by type II fibers. Explosive athletes had ∼30% higher carnosine levels compared to a reference population, whereas it was ∼20% lower than normal in typical endurance athletes.
Muscle carnosine content shows a good reflection of the disciplines of elite track-and-field athletes and is able to distinguish between individual track running distances. The differences between endurance and sprint muscle types is also observed in young talents and former athletes, suggesting this characteristic is genetically determined and can be applied in early talent identification. This quick method provides a valid alternative for the muscle biopsy method. In addition, this technique may also contribute to the diagnosis and monitoring of many conditions and diseases that are characterized by an altered muscle fiber type composition.

Baguet, Audrey, et al. &quot;A new method for non-invasive estimation of human muscle fiber type composition.&quot; PLoS One 6.7 (2011): e21956.