Assuming a smith from the Hellenistic era of ancient Greece was given a lump of metal (specifically a gold alloy) and asked to produce more of the same kind of alloy, what methods would they have available to determine what metals make up that alloy, if any?
Yes, for limited applications.
Archimedes's famous "Eureka!" moment stems from being tasked with determining if a crown was made of pure gold or if some silver had been mixed in. The density of gold was known, so he could confirm the gold's purity from its density. The crown's weight was easy enough to measure, but he needed to know the volume of the very irregularly shaped crown to confirm its density. He got the idea of using water displacement to determine volume and the rest is history. The bit about the bathtub is anecdotal.
Using this idea, a jeweler could potentially reverse this process by forming a hypothesis about what alloys were in use, such as "I think this is a gold and silver alloy" and determine the percentages by checking the density and doing a little math. However, I do not have any evidence this was done.
For those interested, here's the math using modern algebra. The Greeks would have done this using geometry.
The total density of the object is the densities of the individual alloys times their percentages in the object.
d = d1*p1 + d2*p2
The percentages must add up to 100% (assuming only two alloys) so the first percentage can be expressed in terms of the second percentage.
1 = p1 + p2 p1 = 1 - p2 d = d1*(1 - p2) + d2*p2
And a formula to determine the second percentage can be derived.
d = d1 - d1*p2 + d2*p2 d - d1 = d2*p2 - d1*p2 d - d1 = p2(d2 - d1) (d - d1)/(d2 - d1) = p2
Plug in the densities of the object and the densities of your two alloys and you get the percentage of the 2nd alloy. Then the percentage of the 1st alloy is 1 - p2.