For body composition, the diagnostic 'gold standard' is a Dexa scan (AKA 'DXA'), which is a kind of X-ray (lay on a table, scanner goes from one end to the other, kind of like being in a big flatbed scanner - because that's what it pretty much is). This is used more for whole body bone density than body composition, BTW.
There are now body composition *scales*. These use high frequency electrical reflectomotry to gauge composition; most need contact with skin (usually feet, sometimes both hands and feet). Some of those are FDA 'approved' (not sure what that entails exactly). Tanita has some of these that would be appropriate in a doctor's office ranging from $3K to about 13. For athletes, they have 'multi-frequency' scales that can give a readout on intramuscular composition. And they have at least two FDA approved non-commercial models.
Then there are super-inexpensive body composition scales (I got one on sale at amazon for under $15) that are probably OK for seeing trends.
Other ways to measure body composition involve immersion in water (remember that story about Archimedes, his bathtub and a suspect crown? Yeah, it's that.) and a skilled person with a skin caliper judging how much subcutaneous fat you have (but this says nothing about visceral fat, which because of the way it's close to the liver makes it more dangerous).
Finally, there's waist circumference, which *is* a decent readout on visceral fat and neck circumference, which the US military has started using to disambiguate between overweight-because-fat and overweight-because-muscle.
I did some coding work for a drug epidemiology group in the mid-00s. They had access to whats now called the 'UK Biobank'. In that, it seemed that UK physicians had started entering patient waist circumference in the mid-90s (as well as height and weight, which is what BMI is calculated from). I don't recall if BMI was an actual field or calculated on the fly. Anyway, doctors in the UK at least have been moving past the BMI for going on 30 years now.