So you suspect that the disorganized relay of the ECAM messages was due to the PNF's other preoccupation - keeping an eye on the PF.
Yes, that certainly makes sense to me. The PNF was left without leadership but knew something had to be done regarding the ECAM and so tried to do his job but the PF had already launched on his own course of action which left the PNF in a position of not knowing how the PF was receiving the communication. The effort would be halting and interrupted by what the PF was doing with the airplane, leaving the PNF in the lurch about how, where and when to pick up the ECAM discipline and get the operation on the proper track. He had about 40 seconds to do so before the more serious problems occurred and there was no turning back after that point in terms of SOPs.
The appropriate intervention by the PNF in the face of such non-standard actions on the part of the PF would be to hold his own Takeover Priority button down longer than 30 to 40 seconds, (forgotten exactly how long) which locks out the right sidestick in order to fly the airplane uninterrupted and maintain stable flight. Surely some sense of how unstable the airplane was going to get must have entered the PNF's mind.
The PNF then becomes the PF so he could recover the airplane. Once things had calmed down he and the captain could settle the disagreement, but he likely didn't know where it was going until it was too late.