Did the Romans melt down looted golden objects or did they keep them intact?

Warning: answer contains a source in Italian only.

Short answer: I never heard of such a thing like a "general policy" among Roman commanders concerning the usage of looted gold. There were some "moral guidelines", prohibiting for instance looting of temples and shrines, but (as the examples below demonstrate) these were often broken, and each commander acted according to strategic and/or tactical needs. This thesis is supported by many records of different behavior between different commanders; this dissertation thesis offers an example of what I am saying.

Long answer:

But I am wondering if there was a general policy or custom [...]

No, there was no general policy. Depending on the strategic and tactical motivations for an action, the commanders (be Consules, generals or emperors) would decide whether or not, and to what extent:

• to allow the troops looting the city/area. It is in this case very difficult to determine what would happen to any golden artifact: some could be recasted, but some could end up in the hands of people who would keep them for their artistic or religious value
• to collect the goods for them to be shown in a parade in Rome, called "Triumphalia" which celebrated the commander and the army
• to collect the goods for them to be donated to one or more temples, which had a dedicated area called "donarium".

Notice that the three options are not mutually exclusive, in that a lowly soldier could himself elect to donate a looted statue to a temple, rather than selling it.

Two examples: one instance of melted gold and one where it was not (at least not by the authority). In his siege of Athens, Lucius Cornelius Sulla took gold from the nearby temples and had it minted to sustain the war effort. After taking the city he allowed the troops to sack it, to punish the inhabitants for their insolence and set an example against other Roman subjects. Caesar on the contrary

attacked and sacked some towns of the Lusitanians although they did not refuse his terms and opened their gates to him on his arrival. In Gallia he pillaged shrines and temples of the gods filled with offerings, and oftener sacked towns for the sake of plunder than for any fault. In consequence he had more gold than he knew what to do with, and offered it for sale throughout Italia and the provinces at the rate of three thousand sesterces the pound.

(from De Vita Caesarum, Divus Iulius, LIV, Suetonius, emphasis added).

Are there other known examples where the provenance can be traced?

Here things are complicated, because of the repeated pillage of Rome, because of the diffuse looting during decadence times and perhaps most importantly, because of the edict that made Christianity the Roman State Religion under Theodosius.

However, in the thesis I linked, there is a passage that reads:

The Senate then sent the then Consul M. F. Flaccus with a powerful army which laid siege to Vulsinii Veteres until the inhabitants were forced to surrender. The Romans then looted the Etruscan centre; they brought to Rome a whopping 2000 gold and bronze statues, of which a part was exhibited in a "donarium", that Flaccus had erected on the Velabrus, not far from the Capitoline. The Greek historian Metrodorus of Scepsis, cited by Plinius the Elder, who was aware of the episode, accused the Romans of having looted the city for the sole reason of satisfying their craving of the rich offers guarded in the temple. This was confirmed in the '900 when, during an excavation campaign, in the area by the Church of S. Omobono, between the Palatine and the Capitoline, the archeologist Mario Torelli retrieved, studied and published a votive inscription.

Now this does not refer directly to a gold artifact, but according to the text professor Torelli must have found either a gold or bronze statue. More importantly, judging from the tone of such claiming, it must not be an unprecedented finding.

Curiosity. You may or may not know that the Menorah is still sought by the Israelis, who sent representatives to the Vatican claiming that - according to their sources - the artifact is hidden somewhere in Vatican City. The official Vatican answer is however that no such artifact is among their belongings.