Laconian consul Winston Duarte seems to be obsessed with shaping society via strong exercise of executive power, enforced through a military which is held to a strict, unforgiving code of justice. Given this, it seems evident that we're meant to infer that Duarte chose the name Laconia himself, as an explicit reference to the Spartan polis.
The first mention of the name Laconia in the series is in fact after Duarte's defection - in the epilogue of Nemesis Games, we see the crew of a ship that's traveling through the gates as part of Duarte's fleet. The captain thinks of his destination as "the new Laconia Station", suggesting that Laconia is a new establishment.
However, even if we accept these inferences, your other key question remains: How would the people of Earth, Mars, and the outer planets learn of this name?
The means by which Laconia communicates with the rest of humanity is actually mentioned, very briefly. In chapter 13 of Persepolis Rising, Transport Union president Camina Drummer receives a summary of communications with Laconia from an Earth-Mars Coalition liaison:
Lafflin was a thick-faced man with a tight haircut that made him look like a particularly self-satisfied toad. He cleared his throat. "Data on Laconia has always been thin," he said... "The defecting forces from Mars have been playing their keep-away message since before the Transport Union was chartered. They’ve flooded the gate from the realspace side with chatter along the whole electromagnetic spectrum—radio, visible light, X-ray, everything. We’ve had no passive intelligence to speak of. The few times that probes were sent through, they were disabled or destroyed.
This is the first detailed description of outside communications from Laconia, and the first mention of the "keep-away message", but earlier, in chapter 10 of Persepolis Rising, Drummer hears from her aide that the message from the Laconia gate has changed:
"Laconia put out a message, ma’am."
Drummer looked at him. "What?"
"The warning message from Laconia gate was taken down," Vaughn said. "It’s been replaced by a new message. The report from Medina came in"—he looked away and then back to her—"four minutes ago."
In this message, Anton Trejo introduces himself as "Admiral Trejo of the Laconian Naval Command", and announces that the gate is being opened for travel again and that his forces are traveling to Medina Station. Since the Laconian military freely identifies themselves by that name, and since they've been broadcasting a warning message through their gate for many years, it certainly seems likely that Duarte's government identified itself as "Laconia" as part of the warning message or some earlier communication.
However, the waters are muddied by the minimal elaboration on this in Babylon's Ashes. Almost every character who refers to the name "Laconia" in the novel is someone in contact with Duarte's fleet - a member of the Free Navy, or someone aboard Medina Station. When Earth government officials mention it, however, it's used with some ambiguity. In chapter 10, Chrisjen Avasarala asks an aide, "...Do we have anything from our Martian friends about why their rogue navy was so interested in the Laconia gate?" and receives the response, "Not even confirmation that that’s where the breakaway ships went." Later, in chapter 38, Earth Navy Admiral Souther mentions "...the colony systems, including Laconia, where former Martian naval officer Winston Duarte appears to have set up shop."
By the time that the Roci crew and Earth forces assault the rail-gun emplacement on the gate station, everyone is certain that Duarte is indeed through that gate. Things are made further ambiguous by the fact that Duarte evidently hasn't set up his constantly repeating "keep-away message" yet. In chapter 50 of Babylon's Ashes, the Roci examines the gate:
"Anything coming out of Laconia?" he asked.
"We don’t have a repeater on the far side of that gate, but just peeping through the keyhole? Nothing," Naomi said. "No signal. No sign of approaching drives."
So it certainly seems feasible that it was already named Laconia. Alternately, it's possible that as soon as Duarte traveled through the gate, he sent a warning message back establishing his new base of operations with its new name. In that case, the Earth government might not be certain if the message was legitimate, since it's being relayed back to them via sources in the Free Navy-controlled gate space. The oddest thing in this interpretation would be Earth officials using the name Laconia even though it was established so recently. But I think that's the only detail which strongly implies that the name precedes Duarte's defection.