Evaporation is known to change the composition of alloys. For instance if welding Al-Mg by TIG, we use a seam alloy richer in Mg to compensate its evaporation. Al in molten Ti has a similar vapour pressure, in the order of 1kPa.
Vacuum remelting is used to make expensive steel more resilient by evaporating embrittling elements.
Air might shift the amount of N and O in metals. Traces of N2 serve to harden duplex steel while traces of O spoil magnetically soft iron. Traces of N or O embrittle Ti alloys, so welding by TIG apparatus demands a much stronger protection against air than just blowing Ar as for aluminium alloys.
The melting pot uses to add Si to steel. That's so common that we don't even mention Si amounts in steel below an important threshold.
I haven't read about W contamination by TIG welding, but since the electrode is slowly consumed, it must necessarily happen. Traces of W would be undesired in magnetically soft iron.
One relatively clean means to heat metals is induction. Contamination by the melting pot but not by the ultra-hot electrode.
I expect - this is only a guess - that melting pots made of metals with a small vapour pressure contaminate less the melt. That would mean W. Available from Plansee for instance. Cheaper would be ceramics, where oxides of Al and Mg shouldn't hopefully react with Fe, Ni, Co.
My preferred way would melt only locally the analysed alloy. This is done for deposition of Al by evaporation for instance. In microelectronics we used an electron beam for that, but a laser should do it too.
Optical emission spectrometer has the capability to measure alloy composition locally, pointing to precipitates, segregation and do on. Do you want to melt the analysed alloy to get a mean composition? This may demand to exceed widely the temperature at which the alloy begins to melt.