350kW is an amount, like 400bhp, pretty meaningless It needs a time element to be a useful, for example it delivers the 350kW into the battery in 10 minutes.
You have KW (i.e. 1000 Watts) which is a measure of power and KWh-1 (KW per hour, actually often just written as KWh) which is a measure of energy transfer. If you have a car charging at 350KW for 1 hour it will have consumed 350KWh of energy, if charged for 6 minutes it would have consumed 35KWh. So technically a battery should be measured in KWh as it is energy and the charging rate would be measured in KW as it is power.
To be more specific, there will be a charging voltage (Volts being measured as Joules per Coulomb with Joules being a measure of work done and Coulomb being a measure of charge), a charging current (Amps being defined as Coulombs per second) which when you multiply them together gives you Joules per Second which is a measure of the rate of energy transfer (measured in Watts) hence this is how the charger is rated, you them multiple by time and you will get the number of Joules transferred i.e. the energy transferred.