I dont know which hotels have honeymoon suites, but check out here on dubrovnikonline for a list of all the hotels and their contacts.
I suggest Villa Dubrovnik, it is beautiful,very small, very close to the Old Town, yet far enough to have peace of mind http://www.villa-dubrovnik.hr/.Maybe
Excelsior, Argentina or hotel Croatia in Cavtat. Or just choose something cheaper, like hotel Kompas or so. Oooor try the islands, Mljet or Korcula. Whatever you choose, I hope you will be happy that you chose Dubrovnik!