Hedges: Would a heavier gauge of wire be better in this situation? Like 10g? Just asking out of curiosity.
Sure, you could use any gauge, even 4/0 if long enough to provide the desired resistance.
I'm repeating myself here, but in my opinion, the capacitor pre-charge should be part of the inverter design. ... Doing it on the battery can lead to all kinds of weird compatibility issues and edge cases that cause the pre-charge to be completely ineffective.
All we need from the inverter is to not turn on immediately. Either manual precharge or BMS doing it, we want capacitors to reach near full voltage and then switch closed for low resistance before inverter starts sucking multiple amps.
The reason inverter can't easily do this is it doesn't have a controlled switch between DC input and capacitors. Some have a breaker built in, some do not.
BMS has FET or relay between cells and battery cables. Simply turning on FETs gradually so they present high resistance (limiting current) and burn off some heat due to voltage drop (as high as 48V or so) times current. Given enough time, that fully charges capacitors in inverter. Trouble is, inverter will likely draw some current for its internal circuitry, so FET (or separately switched precharge resistor) needs to handle that current. Definitely can't have inverter start producing AC until precharge complete and FET on hard or relay closed.
Best design would be BMS performs precharge and communicates to inverter saying when it is OK to turn on.
Specs for Sunny Island are 4W standby, 25W idle. (Of course, if AC loads present and motors being started, could be 11kW)
4W/48V = 83 mA. Sunny Island powers up in Standby, until you press and hold button to start.
25W/48V = 0.52A, this is what it would draw if AC loads were disconnected and inverter started the moment voltage appeared.
(Problem with other inverters is likely that they power up when DC appears.)
For this inverter, need to design for 83 mA.
Using a wire for precharge resistor, could be a length of telephone or LAN wire, enough for a few ohms or a fraction of an ohm.
That wouldn't be able to power inverter if it turned on before you closed the switch.
Using 12 awg or something larger, enough for about 0.5 ohms, could power inverter up to a few amps (couple hundred watts) with a few volts drop. If switch closed with a few amps and a few volts across it, small inrush. Best to close before that happens.
Don't use a coil of single wire, that is inductance. A coil of 2-wire cable is fine, with the two wires joined at end; current flows out one way through coil and back the other way, cancelling magnetic field and inductance.