I plan to mount one of Ivan's (E1DA) Cosmos ADC's on each meter so the audio cable will only measure a few inches
A few inches? Then you are overthinking all this too much, center pin of the connector to pin 2 on the XLR, shell of the connector to pin 3 of the XLR, shield connected to shell of the connector and pin 1 of the XLR. For a cable of just a few inches you could just use coax cable, and connect pin 3 to pin 1 at the XLR connector. If the buffer output is inverting then switch the pin 2 and 3 connections at the XLR.
Making the cable should take about the same amount of time as reading through all these posts.
output Z is in the order of 5 ohms
25 Ohms per the manual. Should still be low enough, although the Cosmos ADC has a strange input, the impedance is very low for higher gain (or perhaps lower attenuation) settings. Only 640 Ohms input impedance for 1.7V max input, which is really low for modern equipment.
Note that according to the 2203 manual the output is 0.3V for full scale deflection of the meter, so you may need to check your expectation of 1.6V into the ADC (see page 17 of the manual, it is designed to feed weighting filters that accept 1V max signals as input).
Note also that the output of the 2203 is designed to feed through weighting filters that are specified to drive into a 146k Ohm input, so it is not guaranteed that the output buffer is designed to source enough current to drive a 640 Ohm input.
The short of it is you should probably experiment a bit with your setup before expecting you can just connect this to a Cosmos ADC and go.
Do you not have an existing audio interface of some kind you can connect and check? One of those little Focusrite or PreSonus USB connected interfaces would have a more appropriate input impedance, and has adjustable gain so you can check how much gain you need for various sensitivity settings on the SPL meter.