Calculators like this are why I am asking the question.
This one indicates 2.5% drop at 200', which will be no problem. Others indicate 250mcm is needed, or give results that 4/0 may not support 200a even at much shorter distances.
Since you have a meter/main, the house would be fed by a feeder from the pole and not service entrance conductors as defined by code definition.
The NEC allows for an exception for residential service entrance conductors. 4/0 Al is only good for 185A as a feeder but 200A as service entrance conductors. I've never seen a wire size calculator take that exception into account.
So the wire size calculator that returned 250mcm is correct. If you used 250mcm to the house from the pole, it would carry the full 200A @ 2% VD for 125'. The real question is do you need all that?
If you also added to that panel mentioned above and chose a 90A feeder to the garage, 2-2-4-6 AL would work for 130' @ 3% VD. (Total of 5% VD.) Again, do you need all that power?
Guestimate your worst-case simultaneous loads first for both house and garage first.