question

Why are you choosing a 12V system in the first place?

Wire lugs are defined by two sizes:

- wire size (in your case 4/0 - about 120mm^2)

- bolt size.

Lynx Distributor bolts are M8 (about 5/16").

Distance between Lynx Distributor bolts is not indicated in the manual, but from the pictures it seems that center to center is about the same as the busbar width (and this is stated to be 30mm).

I checked the 120mm^2/M8 wire lugs from one manufacturer and the lug width is 28mm. So that one should fit.

You also have to check if the 4/0 wire itself fits.

Regarding your question about the inverter wire size:

I think 3/0 (95mm2) would also be OK for the inverter wires in your case (assuming the wires are not extremely long).

They also want me to use 4/0 to connect my 4x Viltron 200 ah batteries to a Lynx Distributor In.

So that's another Lynx Distributor (with fuses for each battery), right? Not a Lynx Power In?

With battery wires, the thicker and shorter - the better.

With 4 batteries in parallel, you could use thinner wires, but as I wrote, the thicker - the better.

But 3/0 would be more than enough here.

You wrote that "The longest battery cable length to attach the battery terminal to the Lynx In is 5 feet ".

Seems like you plan to use different wire sizes. Do not do that. Make sure that all positive battery wires are the same length and all negative battery wires are also the same length (but positive and negative wire lengths can be different).