How are the boiling points of Tungsten and other metals determined and proved?
According to the 1913 paper THE VAPOR PRESSURE OF METALLIC TUNGSTEN, The Physical Review, vol. II, pp. 329-342:

The vapor pressure (p) of a pure substance is related to its evaporation rate (m) in a vacuum by the following relationship:
$$ m= \sqrt{\frac{M}{2\pi RT}}p$$

Evaporation rates were measured at temperatures in the range from 2400K to 3100K, heating by electric resistance of a filament, and determining the evaporation rate by weighing the filament before and after the evaporation.
The temperature was determined by observing the color and intensity of the light emitted by the filament. (Paper says they will publish a separate paper about the details of the temperature determination technique).
Vapor pressure was extrapolated to a boiling point of 5110 K using the Clausius-Clapeyron relationship.