Justinian didn't think of the entire Gothic War campaign as "reconquering the western empire," because it was all still part of the empire.

He thought of it as "removing an unapproved (and religiously wrong) governor of the western province of Italy." Just like in his mind, the Vandal War was about "removing barbarian squatters from imperial territory."

No claim was ever given up on the west, and that possibility remained theoretically real, if not practically, as in the case when Belisarius negotiated with Gothic forces, and offered to "cede" Britain to them (which hadn't been in functional control of the empire for more 150 years) in exchange for them leaving Italy.

But with that said, it is highly unlikely for him to have wanted to place a competing emperor of equal authority in Italy after conquering it. By then the main reason for the existence of a western emperor, to manage western provinces and military forces (mostly along the Rhine) were gone. So either way, even if Italy had been able to be consolidated, it would've remained a frontier province to Constantinople.

Follow up question: Was there ever a formal Western Roman Empire? The split is usually dated to 395 AD, but did Theodosius I actually decree that the empire be formally split after his death or is that a later narration by historians to make the past easier to manage?