The government chooses not to engage with Gant at all; it never cites the case, presumably to send the message that Gant just isn't on point. Whether that was a savvy tactical move or a dangerous miscalculation should become clear at the oral argument.
The actual Bailey situation is relatively rare, and is one that most law-abiding people are unlikely to encounter. (Mistakes do happen, as evidenced by a 2006 case in which the police held a naked homeowner and his girlfriend at gunpoint in their bedroom until officers realized they had the wrong house. Relying on Summers, the Court found the police acted reasonably.)
But the case may expose Fourth Amendment fault lines that don't track typical liberal/conservative divides. The justices can be placed on a spectrum in terms of the way they handle Fourth Amendment cases. At one end is Justice Scalia, who is eager to rethink Fourth Amendment precedents when they don't square with his understanding of the Amendment's historical purposes. At the other end of the spectrum is Justice Alito, who is generally more concerned with precedent than with original understanding, takes a more forward-looking and pragmatic approach, and is more willing to defer to law enforcement interests.
The contrast between Scalia and Alito has been evident in several recent Fourth Amendment cases. In Gant, the majority followed a course Justice Scalia had charted in an earlier separate opinion, while Justice Alito wrote the principal dissent. So, too, with last term's Fourth Amendment tempest-in-a-teapot, United States v. Jones. The issue in that case was whether placing a GPS tracker on a car constituted a "search" under the Fourth Amendment. All the justices said yes, but they got there in different ways. Justice Scalia's majority opinion looked to the 18th century, concluding that the framers would have thought that the GPS installation was a search because it involved a common-law trespass.
Justice Alito wrote separately, arguing that the Court should stick to the analytical framework developed by 20th-century cases, and openly questioning whether it made any sense to look for historical analogs to GPS technology. "Is it possible to imagine a case in which a constable secreted himself somewhere in a coach and remained there for a period of time in order to monitor the movements of the coach's owner?" Answering his own question, Justice Alito wryly noted that "this would have required either a gigantic coach, a very tiny constable, or both."
Unsurprisingly, Bailey's opening brief is a love letter to Justice Scalia, repeatedly citing one of Scalia's important concurrences and concluding with several pages of argument about the 18th-century background of the Fourth Amendment. The government, for its part, focuses on the practical justifications for allowing seizures like Bailey's. Which strategy prevails should tell us something about just how interested Justice Scalia is in rethinking Fourth Amendment law in his remaining years on the Court -- and whether his school of thought or Justice Alito's has the greater allegiance among the Court's members.