"The empire of the Republic" is certainly one way to refer to the Roman Republic's overseas territory, and it is accurate. After the 2nd Punic War the Roman Republic was undoubtedly in possession of an empire. Therefore referring to the Republic as having or being in possession of an empire would be accurate, but referring to the Republic as actually being an empire would not be as accurate.

Obviously the Roman Republic was not led by an "emperor", or monarch of any type, and the Romans of course still thought of themselves as being a Republic. It also lacked the bureaucratic government typical of most empires. Italy, not that it was referred to as such until much later, still had a republican government. The Senate was still in charge, and most public officials, for lack of a better term, were elected by the people. The overseas territories, however, were viewed as provinces, though not necessarily in an imperial sense, and typically had governors that were appointed by the Senate. The people living in these provinces would have had no say in the things that occurred in said provinces, unlike the Roman citizens that lived in in Rome. But even though most of the people living in these provinces would not have held Roman citizenship, unless they were veterans or colonists, and the provinces themselves did not have their own republican governments, referring to the Republic as a whole as an "empire" would still be inaccurate.

The best comparison for this is likely the Athenian "Empire" from the 5th Century BC. Athens of course had a democracy, and they were the leaders of the Delian League/Athenian Confederacy. Despite their democratic government, however, they acted more like tyrants towards the city-states that were members of the confederacies. In short the Roman Republic, just like Athens, was democracy (or republic) at home, but an empire abroad.