Generally doctors avoid invasive procedures (like using an arterial Cather - aka small tube- to inject dye) or using contrast dye in a CT if there is renal dysfunction. Those specific Dye can injure the kidney.

Step 1: blood tests, urine tests. Blood tests for creatinine estimate renal function. If it is abnormally high, this triggers a doctor to investigate more. Urine test, mainly looking for urine protein. If you are losing protein in the urine, the kidney is not functioning right but the creatinine may still be in the normal range. This does not distinguish bilateral or unilateral kidney dysfunction.

Step 2: ultrasound (possibly with Doppler) -ultrasound shows the size of the kidney. This can easily show if 1 kidney is normal and the other is missing or “atrophic.” If a kidney has not been working well for a while; it will start to atrophy. Then you know that kidney is bad. -Doppler with ultrasound can see if the arterial flow is normal and equal. This avoids contrast which can hurt the kidneys.

That’s usually all you need.

But other tests could include MAG3 scan. A nuclear study. This shows how much function is in each kidney. This would definitely show if 1 kidney was failing earlier than another

https://en.m.wikipedia.org/wiki/Radioisotope_renography

Kidney biopsy. Usually though you do not biopsy 2 kidneys. As one reply mentioned it is more common to have kidney damage to both kidneys. Because both kidneys do the same work, they are affected similarly by conditions such as diabetes and high blood pressure that stress the kidney over time.

If 1 kidney was not working well, typically it is congenital, has bad blood flow, or had a blockage that does not allow urine the leave that kidney.