Pardon the lateness of this reply. I did not see this post until recently.

Your question, to my knowledge, has not yet been posed. Thus, the research presented below was undertaken for the direct purpose of answering this question.

It is no secret that the system of the republic did not simply vanish during the reign of Augustus. The early empire, often called the ‘Principate’ did not quash the republic, it was adapted from it. The veneer of the Republic remained until, most would say, the reign of Emperor Domitian at which point it was quashed. This, however, does not mean to say that the pre-Augustan Republic, the post-Augustan Republic and the ‘Republics’ of Domitian, Hadrian and even that of Severus were the same. Times change and so must the mechanisms of government. Potter (2004), I think, discusses this well. Read his preface (xi-xiv) for a brief overview.

That said, Jordanes is not a contemporary of Decius. Writing in the 6th century (ca. 551, about 300 years after Decius: see my other post on his origins: https://www.reddit.com/r/AskHistorians/comments/8r06ph/why_cant_we_pinpoint_jordanes_origin_in_the/ ), Jordanes’ perspective is entirely anachronistic (or ‘etic,’ if you will). Our sources for the 3rd century, for the sake of comparison, are poor. Much of what we have contemporary to the 3rd century comes from the most problematic Historia Augusta. Thus, the real answer to your question about whether a 3rd century soldier truly saw his system as being either a Republic or an Empire is factually unknowable. But, as Potter shows, the traditions of the Republic maintained until the system was completely overhauled by Diocletian and then again by Constantine (ibid). Therefore, it is likely that some thought the Republic was still alive but there was little question who was in charge by the 3rd century: it was the Imperator, not the senate.

It is also worth noting that the Historia Augusta, troublesome as it is, does refer to the state as a ‘Rei publicae,’ an identification that is not always translated into English (see, in Latin, Historia Augusta, Hadrianvs, 24.2, 11; Helivs, 3.8, 4.5-6, 5.2, 6.3, 7.2; Antoninvs Pivs, 4.3, 7.5, 9, etc; Marvus Antoninvs Philosophvs, 2.9, etc; Severvs, 19.6, 20.1, etc; Gordiani Tres, 19.6, 22.1, etc; Tyranni Triginta, 1.1, 5.4, 5.5, 9.1-2, 10.10, 15-17, etc; Carvs et Carinvs et Numerianvs, 1.1, 4, 2.1-2, 4, etc). Certainly more examples can be added, but this should suffice. Written in the late third/early 4th century, Historia Augusta might have been based largely on information from Dexippus (a contemporary 3rd century author; Potter, 1990, 365-369; 1999, 199-200, n.82; 2004, 234, n.79; cf. Barnes 1991, 1-28) and/or Nichomachus Flavianus/Eunpaius (former late fourth century, latter late fourth to early fifth; Paschoud 1975, 93-99). Therefore, it is certainly possible that many still regarded the system as a republic, especially if Historia Augusta is indeed citing Dexippus.

Jordanes himself, in Romana, frequently referred to the emperors as having lived in the Republic (Caligula, 261; Nerva, 266; Hadrian, 269; Antoninus Pius, 271; Julian the Apostate, 304; Gratian, 315; Theodosius, 318, 321; Zeno, 342, 346 even after the western empire became governed by Gothic kings, 352, 355; Anastasius, 357; Justin, 359, 362; Justinian, 366, 375, 377, 388). This might have been to continue Marcellinus Comes’ argument against Zosimus that Christianity was not the cause of the decline of the state (Croke, 1983, 81-119; cf. Marcellinus Comes 476.2, 489); instead, even after the transition of the west to Gothic kings, the eastern empire was still the very same system passed on from Caesar and Octavian Augustus, uninterrupted. This would mean, arguably, the eastern empire could re-instate the Republic in the west if it reconquered it…

That said, and this will really answer your question in an obvious way, the last emperor Jordanes indirectly calls a Republican prince, if you will, is Antonius Pious (ca. 138-161 AD; Romana, 271). After that, the word ‘republic’ is not mentioned until Emperor Julian (ca. 361-3 AD; Romana, 304). A fairly long stretch of time, though there is a lacuna (a portion of the text missing) in line 302. Even so, note Jordanes' remarks on Diocletian (ca. 285-305 AD):

“So then the tribe of the Carpi was defeated and transferred onto Roman soil. Moreover at that time Diocletian, as the first of all emperors, ordered that he be adored as a god, and braided gemstones in his clothes and footwear and a diadem on his head, whereas before him they had all had only a purple cloak to distinguish them from private citizens, and were saluted as other magistrates” (Romana, 299; translated by Regan, date unknown: http://www.harbornet.com/folks/theedrich/Goths/Romana.htm).

In Jordanes’ eyes, the Republic continues after Diocletian, but he is the first emperor whereas before him, the princes (who we understand as emperors) were regarded as magistrates.