I suggest a tour through Bosnia-Herzegovina. Begin with a night in Split and explore Diocletian’s magnificent third-century Roman palace, a Unesco World Heritage site, and then take a three-day road trip inland through the mountains.
Drive from Split to Sarajevo (180 miles, five hours), passing through Trilj and continuing over the border through Livno and Bugonjo to Donji Vakuf. Here you could take the side-road to Jajce (waterfalls and medieval fortress). Continue to Travnik (Ottoman architecture) and stay in Sarajevo. Surrounded by dramatic mountains, the Bosnian capital is home to many mosques and churches. The effects of the war are still visible, but a fair degree of cultural tolerance has been restored.
The next afternoon, drive from Sarajevo to Mostar (90 miles, two hours), a scenic road passing through Konjic and Jablanica. Stay the night in Mostar and see the Ottoman old town, which is based around the reconstructed “old bridge”.
The next day, drive from Mostar to Dubrovnik (90 miles, two hours 30 minutes), following the emerald-green River Neretva down to the sea, then tracing the coast south east. You’ll have earned two restful nights in Dubrovnik, and you should save Montenegro for next time.
I’ve checked and there are no restrictions on taking a hired car into Bosnia from Croatia. The Foreign and Commonwealth Office does warn against the risks of old landmines in Bosnia-Herzegovina, so take care away from main roads.