The class A CDL would cover you, as in Texas for the non-commercial (exempt) driver licenses the higher classes (A, in your case) cover all lower classes (B & C) except class M (motorcycle).
It's the same principle as the class A CDL covering your requirement for a non-commercial class C to drive your private car.
The downside, as you well know, is that any traffic violations, even in your private vehicles, count against your class A CDL.
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