I am not sure how Tesla prep their cars for Ludicrous Plus mode. What I remember back in the days was there are electric motors we asked manufacturer to provide us that would bump the power output for 1/3 and it would be ok to do that for 8 seconds before it might cause problem (demagnetize). It is definitely some prep work in cooling and heating (keeping each component in its optimal temperature range) as well as flushing out some latent heat inside the cooling system (speed up cooling pump and fan and pre-cool or pre-heat all the coolant).It's a relatively minor thing to discharge quickly. The biggest issue is always going to be heat generated, and I wouldn't necessarily be that concerned with a good battery cooling system.
I didn't actually run it in that mode, but I looked at Ludicrous Plus mode when driving a Model S P100D loaner. It said that it would need to prepare the battery for it, which I believe involves heating it to allow for faster discharge. But that's also the case when prepping for Level 3 DC charging, where allowing navigation to select or know the charging destination gives the system time to prep it.
The limiting factor for the power discharge rate would likely be in the tire first, then batteries, then inverter and converters, then electric motors.