You seem to be aware of the meaning of res publica as something much closer to English "state" than "republic," so I won't waste anyone's time with that. Res publica was not synonymous with civitas, which means something more like "citizenry" but by extension could mean the state as well. A state could have civitas (in fact, it must to be any political entity at all), but it was not necessarily res publica--whether res publica populi Romani or even regalis res publica--if it lacked the necessary degree of organization. So, Caesar refers to the Gallic communities as civitates but not as res publicae. That the term was not necessarily applied exclusively to the highest order of unified hegemonic state, no matter what its precise organization, is however very clear. Quite frequently the term applied specifically to city governments. Festus records a note from a late Republican jurist who says that:

initio fuisse, qui ea conditione cives fuissent, ut semper rem publicam separatim a populo Romano haberent, Cumanos, Acerranos, Atellanos, qui aeque <cives Romani erant et in legione merebant, sed dignitates non capiebant.>

initially there were those who were made citizens [of Rome] on the condition that always they might maintain their own res publica separately from the populus Romanus, e.g. the Cumani, Acerrani, and Atellani [Cumae, Acerrae, and Atella] who were equal Roman citizens and served in the legions [or, arguably more properly, "in the levy"] but did not maintain their own magistracies

To this list we might add some others, since a little later Festus says:

Praefecturae eae appellabantur in Italia, in quibus et ius dicebatur, et nundinae agebantur; et erat quaedam earum R. P., neque tamen magistratus suos habebant. in †qua his† legibus praefecti mittebantur quotannis qui ius dicerent. Quarum genera fuerunt duo: alterum, in quas solebant ire praefecti quattuor †viginti sex virum nu pro† populi suffragio creati erant, in haec oppida: Capuam, Cumas, Casilinum, Volturnum, Liternum, Puteolos, Acerras, Suessulam, Atellam, Calatium: alterum, in quas ibant, quos praetor urbanus quotannis in quaeque loca miserat legibus, ut Fundos, Formias, Caere, Venafrum, Allifas, Privernum, Anagniam, Frusinonem, Reate, Saturniam, Nursiam, Arpinum, aliaque conplura.

They were called prefectures in Italy, in which justice was dispensed and market days held, and they had a certain kind of res publica, though still they could not maintain their own magistracies. According to the laws [the text is corrupt here] prefects were sent yearly who dispensed justice. There were two kinds of these cities: one in which four prefects were accustomed to go [the text is corrupt again, something about twenty and six and nu is just not a fucking word] were created by public vote, in these cities: Capua, Cumae, Casilinum, Volturnum, Liternum, Puteoli, Acerrae, Suessula, Atella, Calatium. The other into which those whom the urban praetor every year had sent according to the laws of each city would go, as was the case with Fundi, Formiae, Caere, Venafrum, Allifae, Privernum, Anagnia, Frusino, Reate, Saturnia, Nursia, Arpinum

Festus is relatively late (second century) in comparison with what we're talking about, but his de Verborum Significatu is an epitome of a grammatical work by Flaccus of the same title, written around Augustus' lifetime. Moreover it's pretty clear from inscriptions and more contemporary texts that res publica did not apply only to the sort of umbrella "Roman" res publica, but that little res publicae could exist within Italy and even in the provinces. In the first quotation from Festus he's talking about specifically municipia, those towns which originally had usually some form of Roman citizenship sine suffragio but which following the Social War were communities of full Roman citizens. They were not, however, coloniae, and thus maintained local political identities left over from their days as allies--and, in some sense since they technically maintained local administrative bodies, they did hold some small degree of localized autonomy. By Augustus' day municipia were generally understood as Italian communities that were simply not coloniae, although many were refounded as such.

Naples was one of the earliest Greek-speaking allies the Romans had. They minted Rome's first coinage and proudly displayed their loyalty during the war with Hannibal. Despite close contact with the Romans Naples remained an allied city for rather a long time, until in 89 it was finally made a municipium. Augustus resettled the city as a colony, but the city maintained a decidedly foreign character well into the Principate, partly because of its privileged status in the Republic as one of the more valuable allies, but also because of the strength of its culture and the way that Greek culture in the city was encouraged by Roman landowners who dwelt in the summer months in resort towns along the bay of Naples (like Pompeii). Thus, the city maintained Greek magistracies and was still setting up Greek inscriptions when Caracalla extended citizenship to all free provincial males. So while I do not know off the top of my head the precise legal status of the Neapolitans and the extent of their local territorial jurisdiction, Suetonius' description of them as res publica is not at all out of character