I have absolutely no source so this could be utter rubbish, but for some reason I have in my head a reason for HMQ’s favouritism of PA.
PA was born 10 years after Anne. During that time PP had been rather a cad and the relationship between him and HMQ was not at its best, despite HMQ being very much in love with PP.
There was however a rapprochement and PP stopped being such a bugger (or at least hid it better) and that resulted in PA and PE being conceived.
The reason HMQ always favoured PA was related to the happiness she felt when his was born due to her and PP being in a second honeymoon phase.
She was also far more hands on with PA and PE than her eldest two, so accepting flaws in character would also cut closer to the bone.
So essentially it was more about what feelings he elicited rather than him as a person and acknowledging what a prick he was would have required an emotional readjustment HMQ was unable/unwilling to do.
To be fair I think it must be hard for any parent to admit they’d borne and raised a bad’un and I do wonder how much HMQ was (ironically) “spared” of his behaviours by coutiers who tried to contain him as well as they were able, rather than be frank about his actions,