Battery bank paralleling
Yes, but it won't be pretty. 

Big damn cable

This is not that weird.  Tram, trolleybus, streetcar and light rail systems distribute more power farther.  

This stuff will be aluminum, because copper makes no sense at these sizes. (1000kcmil Cu cost $10/foot last time I bought it).  

Plug into a voltage drop calculator and you see what kind of voltage drop you'll have at what amperage.  For instance, assuming 1000kcmil Al wire (the largest readily available insulated wire in the US) you'll have about 4 volts drop at 20 amps, round trip.  

Pole line?  Consider actual power line wire.  This is uninsulated, but is enormous, much larger than 1000 kcmil.  You may be able to get some used, but it'll be a bugbear to move. The stuff is heavy.  

Pump up the voltage

Pump the transmission voltage up to the insulation limits of the cable, allowing much smaller cable.   If you ever need to switch directions, it will require some active battery monitoring and intelligence to make decisions which way to send the power.  

This may also auto-correct for voltage drop.   There are two ways to do this: 

DC-DC.   This involves  fairly exotic high voltage DC-DC converters.  This also has some practical limits on maximum voltage. And playing with high voltage DC which is dangerous stuff.  750V is a common voltage used by trams, 1500V by interurbans.  

DC-AC-DC.   This involves a common off-the-shelf inverter to create 120V or 230V AC, then a common transformer (or two) to kick the voltage up to 480, 600, 2400 or 12,000 volts AC, transmit at that voltage.  Then another transformer to knock it down to 120V or 230V. Then a common off-the-shelf battery charger. 

How do you plan to get it there?

A lot of it boils down to how you intend to wheel the power 4km. This is a distance where a pole line makes sense, and nice thing about a pole line is you can easily run very high voltage, and with the the highest voltage transformers safely up on the pole top.    

However a pole line requires that you go out and clear trees and brush which threaten the wire.   It helps to own an bucket truck.  

If you go underground, maintenance is lower, but voltage is limited unless you use exotic wires with extreme insulation.   Whatever size wire reels come on, it helps to have a wiring vault at that spacing.

Copper thieves

With such a length, you are vulnerable to thieves tearing out your wires to sell to metal scrappers.  They can easily get at wire which is on a pole line or in conduit.   They will have more trouble stealing direct-burial cable.