Even with a second pulse, the Barak 8 would not be able to hit a target 100KM away. As for the ER type, I don't think India's seven destroyers would be able to accommodate a Barak 8 with a huge booster without a change in the launcher

Gen Next missile defence shield built by Israel and India clears first hurdle | India News - Times of India
India News: Fourteen Indian warships, including aircraft carrier INS Viraat, are currently equipped with the Barak-I system. The new LR-SAM system with Barak-8 in

By the way, I was wrong about the minimum range, it's actually 300 meters, not 500.
Barak 8 launcher can handle the booster. The booster is just 0.09 meters more in diameter.
At the very least, China and the United States can maintain an air defense net of at least 500 kilometers for the entire carrier fleet
That is to say, India's current destroyer can only maintain 200 kilometers of air defense network, of course, this is enough in the Indian Ocean
The Americans have different missiles for different roles. While India has not bothered about long range SAMs on ships for now, that will come later, the Barak 8 provides sufficient long range and even point defense, but the same on US ships is provided by RIM-162 ESSM, not SM-2/6.
So even their defenses are designed like S-400. But I have only seen Chinese ships with HHQ-9 and HHQ-10, similar in structure to what the US had in the 1990s with just RIM-116 and SM-2, and back then they also had the older Sea Sparrow.
Even for the Americans, a range greater than 200 or 300 km is pointless, forget 500 km, because the main threats fly below 8000 meters. The real threat is well within 100 km. SM-6's claimed range is 240 km but potential range is 370-400 km. That's enough to cater to threats at altitudes necessary for firing missiles.
500 km is just the radar horizon, the max potential for a ship using its own sensors. If you have aircraft to provide targeting, then you can even have a 1000 km SAM.
China's HHQ9 is an active radar-guided missile, which is a basic common sense
It's only an assumption made for the later model HHQ-9B.