Can battery inverters output reactive power?
First, Power Factor (and its correction) does not relate to the batteries themselves in any way. Power Factor has meaning in AC circuits and the batteries are DC devices.

The batteries are never discharged by the charger. They are discharged by the equipment they power.

One of the problems discussed in the paper is that the power factor of the charger may depend on the charger's load. It is implied that the charger's load depends on the battery "state of charge" (or discharge depth).

That's where the power factor and the discharge are related (not much, in fact).

p.s. 

a traditional, heavy, iron-transformer-based charger looks like this:
AC grid voltage => Transformer => AC battery voltage => rectifier => DC battery voltage => battery

The source of the reactive energy (and low PF) is the transformer and/or regulating circuit that may be at any of the above stages.

a modern, inverter-based charger looks like this:

AC grid voltage => rectifier => DC grid voltage => PFC => DC above-grid voltage => high-frequency inverter => some AC => small transformer => AC battery voltage => rectifier => DC battery voltage => battery

Here, the main PF offender is the first rectifier. The PFC may or may not be present and it may or may not be equally efficient at all possible power levels.