It was allowed to exist as a theoretical fief of the Ottoman Empire, so if it had survived the Napoleonic Wars, or been restored, then during the 19th century it may well have broken away completely in the wake of Montenegro.
Montenegro of course is very close and IIRC was rather covetous
This is probably true. Getting it through the Napoleonic wars wouldn't be too difficult. One of the main motivations for the French invasion of Dalmatia was to counter the Russians who had seized the Ionian Islands and Kotor. So if the Russians stay out of the Adriatic so do the French, probably. It would be easier still if Austria somehow retained Dalmatia, which would probably require them to willingly subscribe to the continental system. Then we can preempt the formation of the Illyrian states and the birth of pan-south-slavic nationalism. At the very least it has to remain unoccupied because no Republic, Ragusa, Lucca, Genoa nor Venice were restored IOTL and its hard to see how Ragusa could have been once it had been annexed to a major power.
Ragusa was also in a better position than Venice in some respects. It wasn't quite as ossified institutionally and economically it was still doing well playing off its neutral status in the Med, its ability to trade into Ottoman territories and Catholic ones alike. They had one of the most astute diplomatic corps in southern Europe. Probably its biggest issue would have been nationalism. The Ragusan patriciate had a rather fluid identity slipping between Slavic/Croatian and Italian. This could have worked for or against them. I think that the traditional institutions of the Republic, the Senate, Rector, St Blaise etc could have provided a solid foundation to confront 19th century nationalism especially sense we're considering them at a civic rather and than national scale.
In terms of regional threats I think Montenegro was more interested in Albania Veneta (Cattaro, Budua the Gulf of Kotor basically) than in Dubrovnik. Serbia and Croatia are only threats as much as they represent a South Slavic state otherwise they cancel each other out. Yet another reason why limiting pan-slavic nationalism would be helpful. Preventing the formation of Yugoslavia would probably be the best way to secure Ragusan independence unto the present day. Though regardless until the tourist industry takes off the Republic is likely to be rather poor unless it can somehow capitalize on the Suez Canal, doubtful for such a small state. There's no real room for industry or modern shipyards and once railroads penetrate the Balkans its role in trade there is finished.