There's more evidence than simply Catherine's memoirs. Specifically, Elizabeth's actions and reactions. She was no idiot when it came to Peter and his inability to fulfill his relationship with Catherine. She also detested him and probably took pity on Catherine. Any female ruler is in peril when she lacks a legitimate heir, Peter was not truly legitimate because he was not Russian born. But "his" son, born and raised in Russia by the Empress, would have served to secure not only Elizabeth's position but also Peter's. And Elizabeth probably realized this needed child would not biologically come from Peter. Besides, even at that time Catherine was considered a better potential ruler than her husband. Any child that came from her would have been valued. Furthermore, Elizabeth had an iron grip on her court and kept an absurdly close eye on Peter and Catherine. She (and indeed the whole court) would have known if Catherine was sleeping around and if Elizabeth had been displeased (or had she wanted an heir only from Peter), she undoubtedly would have continued locking Peter and Catherine together. The fact that she removed her spies from Catherine's circle and stopped forcing Peter and Catherine together is proof, in my opinion, that she approved of the adultery. Remember, too, that Elizabeth was highly strung and not afraid to punish those who went against her. If Catherine had been having an affair Elizabeth considered illicit or damaging, Catherine would undoubtedly have incurred the royal wrath and paid heavily.
As for Elizabeth's pride in her Romanov heritage, while she loved being the daughter of Peter III if she had any sincere intention of preserving her father's line she would have married and produced her own heir instead of plucking a distant relative out of obscurity. She willingly and knowingly did not produce her own biological heir. By the time Catherine started sleeping with Saltykov, Elizabeth probably realized the true Romanov line would end with her because of her poor decision - making Peter her heir. It would have been enough for her to have been the last blood Romanov (she probably got some kicks and pride out of it), because in spirit the Romanov line would always continue. And that seems to have mattered most.
And then look at Peter's actions. He never struck out at any of Catherine's lovers, never sought to humiliate them or have them removed (at least until they had served their purpose and gotten him another heir). He could even be friendly with them, and indeed they were good and well liked men regardless of rank (some were even quite popular, much more so than Peter). And Catherine never flung them in Peter's face, never actively sought to cuckold him. Catherine and Peter seemed to have worked out a truce, realizing that while they could not have children together, they could at least create a makeshift, stable family for the sake of the empire.
There were no secrets in the Russian Court. Everyone would have known Saltykov was Paul's father. But there was no opposition or scandal. Only acceptance.