Is there a museum in Dubrovnik or Split that provides a more-or-less "neutral" exhibit of the tragic war of the 90s? I realize this is asking for something near impossible (as history of any war is almost by definition not neutral--particularly a recent war), but I thought it was worth asking.
There is a small one at the top of Mt. Srd. Take the cable car up and down. There was also a small museum called the Memorial Room of the Defenders of Dubrovnik.
Yeah, "neutral" is asking a lot for a war that ended so recently. There are several displays in Sarajevo which involve things that happen in the Siege of Sarajevo, but they are pretty clearly slanted. It's best to go to such displays with an understanding of the war, as much as anyone outside of FRY can understand it.
Split was unaffected by the war directly and although Dubrovnik suffered from shelling in the beginning of the war, but most of the tragic war took place in Eastern Slavonia(Vukovar, near Osijek) which is as far from the Adriatic Sea as it gets and very few travellers include that part of the country in their itinerary.
The Red Fico monument in Osijek is really cool. Take a look here, since you'll probably never go to Osijek: