Grimy,
The CA Commercial Drivers Handbook (on the CA DMV site) current issue is the 2017-2018 version. It has been revised since I was looking up requirements in 2014.
It is not as clearly defined as I remember it being in 2014.
But if you look at the sheet where they have columns headings "You May Drive", "And You May Tow" and "Examples" (the Examples column has pic-to-grams).
The only 3 axle trailer example in the pic-to-grams is listed with Class A Commercial License. That truck/trailer combination is a large 18 wheeler type rig.
I will be the first to admit that as listed in the current Commercial Drivers Handbook it could be subject to interpretation.
I would think that if you didn't have a commercial Class A and CHP wanted to issue citation it would be in an uphill battle with that.
As a side note, I find it interesting that in CA (even on interstate) for any vehicle pulling a trailer the speed limit is 55 mph.
My observation was that I appeared to be the only one that was adhering to that law.