First, the AHJ is wrong in the scenario in the OP for the various other reasons discussed. But I wanted to address the above statement.
(Almost) nothing in Article 220 addresses continuous vs non-continuous. Whereas other articles refer to the continuous vs the non-continuous load, in particular 215.2 and 215.3. So while Article 220 spits out a single number for a load, since other articles are asking for a break down of that load into continuous vs non-continuous, when applying Article 220 you need to track the inputs to determine what portion of the answer is a continuous load. This should really be made explicit in Article 220, right now it is only implicit.
So the upshot is that if your 220.83 calculation comes out to 138A, you also need to note that 48A of that (*) is a continuous load, and use that when applying 215.3 (the service OCPD is also the OCPD for the feeder supplying all the loads) and 230.42(A)(1).
Cheers, Wayne
(*) Actually, I don't think it's 48A because of the 40% factor in 220.83. It's somewhere between (8kVA/240V) + 40% * (48A - 8kVA/240V) = 39.2A and 40% * 48A = 19.2A, e.g. it could be (48A / total connected load) * 138A. It's problematic that Article 220 doesn't provide us any guidance on how to compute the portion of the load that is continuous. But even if it is 48A, you're fine with a 200A service with 4/0 Al conductors, 138 + 25% * 48 = 150A.