It is necessary only to calculate the Primary General Membrane stress (Pm), and not even the Primary Local Membrane Stress, and compare it to the limit in 4.1.6.2. If you have Primary Bending, then you will also calculate that, and compare it to the limit in 4.1.6.2.
However, in almost every situation, Pm is a hand-calc, so you will not likely use FEA for that.
And yes, internal pressure and pressure thrust will indeed generate secondary stresses, which have no limit when considering the test condition. Secondary stresses also have no limit when considering the failure mode of Protection Against Plastic Collapse, either.