You talked about a battalion, I gave you a drawdown on requirement of saturating 8 radars.
Also remember that attack of all those 296 missile need to be "simultaneous", and you would need much more than 50 to keep Pantsirs busy. If the attack come in waves, it would be easily dealt with by Pantsir and S-400s combo.
Also remeber that 40N6 is active radar homing (thus not needed target to be engaged by Radar) and rest of missiles are SARH which need Radar to paint their target when the reach close to their target. Unless all of your 500 or so missiles are closing in on S-400 simultaneously, it could take turns to take out those in lead.
There is no outsaturating a system like S-400. Only way to take it down is the one stated in that Raython's video that you have posted.
India is buying 6000 missiles which could arm upto 16 regiments (going by Russian classification). At least some of these would be deployed close to Pakistan's border.
S-400's longest range missile (400Km range:40N6) could not target fighter jets. It was designed to take out AWACS, Bombers, Tankers, and EW planes. Some batteries of S-40 would be stationed for purpose of sniping out your AWACS so that IAF could have free run after PAF is blinded.