I expect that for aluminum wire, the cost difference between 2/0, 3/0 and 4/0 will be low. I hear PVC conduit is expensive these days, not sure how big a difference 2" vs 3" would be (there's also 2-1/2"). You can price out the various options. Also check on the equipment specs on both ends, whether the lugs provided can accept 4/0 wire (if not, you can splice to a smaller size at each end, but that's some trouble). How would you provide a 150A supply for this feeder? In the few panels I'm familiar with, 2 position double pole breakers max out at 125A, so 150A would require a special 4 position double pole breaker.
As to sizing the EGC, it works like this: in aluminum, a #6 EGC is good for up to 100A, and a #4 EGC is good for up to 200A. However, this assumes that your circuit conductors are the minimum size allowed for the feeder. I.e. for a 100A breaker, #1 Al; for a 125A breaker, 1/0 Al for a load up to 120A, or 2/0 Al for a load of 120.5A to 125A; for a 150A breaker, 2/0 Al for a load of up to 135A, or 3/0 Al for a load of 135.5A to 150A. [That's all assuming 75C wires and no derating required for temperature or number of conductors.]
When your circuit conductors are upsized, the EGC should be upsized correspondingly. So if you put 1/0 Al on a 100A breaker, you should use a #5 Al EGC (which isn't commonly available, so use #4). Or if you put 4/0 Al on a 125A breaker, you need to use a #2 Al EGC. [There is an argument that for a calculated load of 120A or less, you would need a #1 Al EGC, but perhaps that's taking things too far, I'm not clear on this subtlety.]
This rule is found in the NEC as 250.122(B). The wording on that section has changed some in the past decade, so depending on which version of the NEC you are under (2014, 2017, or 2020), it may be that the EGC upsizing only applies when the circuit conductors are upsized for voltage drop, rather than in all cases. So if you have a compelling reason not to upsize the EGC (e.g. it would require bumping up one conduit size and you don't want to), and if you are under a version of the NEC that only requires it for voltage drop, and if the minimum size checks out for voltage drop and you are upsizing the circuit conductors for some other reason, then you could skip upsizing the EGC and just use #6 or #4 per the OCPD size.
Cheers, Wayne