The basilica Neptuni was erected by Agrippa in 25 B.C. to commemorate the naval victories of Augustus. Although called a basilica, it appears to have been a temple dedicated to Neptune, standing in the middle of the enclosure formed by the porticus Argonautarum. It was burned in 80 A.D., and restored, probably by Hadrian. A drawing of the sixteenth century represents the building as hexastyle and peripteral, with fifteen columns on each side. A part of the northeast side is still standing, although much injured by fire, and forms part of the building that was until recently the Custom House, but is now the Bourse. This part consists of eleven fluted columns of white marble with Corinthian capitals, and a richly decorated entablature. The columns are 13.10 meters in heiglit and 1.44 in diameter, and date from Hadrian's restoration. The cornice has been so badly restored as to appear now in three patterns. The wall of the cella behind the columns is of peperino, and the original marble lining has entirely disappeared. Cella and columns stand on a lofty stylobate, which is now buried beneath the surface of the ground. This stylobate was adorned with reliefs, - those beneath the columns representing the provinces, and those in the intercolumnar spaces trophies of victory. In all, sixteen statues of provinces and six trophies have been found, but they are scattered in five different collections in Rome and Naples.
An exact restoration of this building is perhaps impossible, but it is quite probable that the temple was octostyle, peripteral, with fifteen columns on a side. If we suppose that a wide flight of steps occupied the whole front of the stylobate, there would be space for thirty-six reliefs beneath the remaining columns of the peristyle, and this is exactly the number of the provinces in the time of Hadrian. Other remains have been found, among them pieces of travertine pavement within the cella, and fragments of columns and cornice and a granite threshold 4 meters below the present floor level.
Pictorial Dictionary notes
Basilica Neptum, was between the Pantheon and the Porticus Septa Julia, Piranesi moves it to the other side of the Via Lata. It was badly injured in the fire under Titus in 80 AD; it was restored by Hadrian probably in conjunction with the reconstruction of the Pantheon; there is an existing drawing by Palladio and a fantasy reconstruction by Lanciani.
Re: research assistance
The Basilica Neptuni seems to be a building that Piranesi 'played' with within the Ichnographia. According to Nash's Pictorial Dictionary, Palladio drew a plan of the Basilica Neptuni (as illustrated therein), and Piranesi obviously knew this plan because he used it [and labeled it Xystus] within the Ichnographia. In my experience, whenever Piranesi makes what seems to be an obvious mistake within the Ichnographia, that's usually a sign that Piranesi is playing a(n inversion) game. That may not be the case here, but you should at least be aware of the (intentional?) transposition of buildings.
Platner also has some to say about the Basilica Neptuni, as I'm sure other sources do as well.
From the 'Catalogo' of the Campo Marzio:
Basilica di Nettuno, «El Sparzian. in Adriano» Vegessi Portico di Nettuno. --I assume this is a reference to the Hadrian biography within the Historia Augustus.