I'm frankly not entirely sure where the Elgin Marbles were in the Dollhouse, nor who the 'client' was. But, if it was indeed Greece, and they were being stolen from the Britsh Museum (where they actually currently are) and given the ongoing controversy surrounding the fact that Britain has them and Greece doesn't -- well.. I'm sure if Greece were willing to buy them back for a fair market value the British Museum would be happy to return them, especially if a deal to have them return to Britain for exhibitions from time to time could be worked out...
(Meanwhile, if Greece just goes and steals them back that's going to be rather awkward to explain...)
I rather liked the use of the Parthenon marbles (but then, IAAAncientHistorian). There was even a line about "that's one of the metopes, we want the frieze", which was a nice bit of geekery (they are two separate parts of the architectural decoration). But a lot of the explanation was done at speed and in passing. Fine for a geek reference, but maybe not for something that's relatively important to the plot. Whedon/the writers didn't seem sure whether or not he/they wanted it to be a major part of the episode's plot, or just a MacGuffin.
The missing explanation was really simple: there's more bits of the Parthenon around than the bits taken by Elgin. It's one of those monuments where to see all of the surviving sculpture requires a trip to a number of museums (the British Museum, Athens, and 9 others, thanks Wikipedia), and some of it isn't on public display. So the idea that there would be one bit of the frieze that was in circulation makes sense.
As for buying them back, these were black-market artworks, remember? They could, of course, have used law enforcement - Interpol takes these things rather seriously - but given how impressive the "collection" in the vault would have been, I can imagine that not being possible. As for stealing them back, I don't imagine that a Greek politician would be in too much trouble for recovering a piece of national heritage from criminals by staging a daring raid on an international art thief (IF they got the bit back, of course). At least, a Greek politician within a tv-show universe, and possibly IRL as well given how much skullduggery there is on even the legitimate antiquities market.
