I could not imagine that whatever this process must have looked like in the 1840s was completely bilateral for all parties. This was the era where you just sent an archaeologist and got some local day laborers to dig where you heard there might be good stuff, and took whatever you wanted back no questions asked.
> I could not imagine that whatever this process must have looked like in the 1840s was completely bilateral for all parties
Then let me help your imagination: A rich Bavarian king frequently goes on holiday in Italy, as was the fashion in Europe at the time. There, he regularly meets and intermingles with rich Italians who share his interests in antiquities. He purchases numerous artifacts from them, at a price both agree is fair.
It would be great if it happened like that, but all the museum has produced in terms of provenance is that it sat in their inventory at one point in the 1800s. No mention of the circumstances of how it came to be in the collection, unlike the legitimate purchaser who bought it and has a receipt.
Roman artefacts are dug up all over western Europe and even further. Is your premise that only one country is the allowed descendants of the Empire and that other constituted parts of the Empire aren't allowed their shared legacy?