Cooper had already told local media of his exclusion from the 30-strong party but coach Robbie Deans said missing the camp did not rule a player out of participation in the three Tests against the tourists in June and July.
Former rugby league international back Folau was included on the back of improved performances for the New South Wales Waratahs in his debut season in union after making the switch from the Australian rules code.
Folau was one of seven uncapped players invited to Sydney to participate in the camp on Sunday and Monday, when they will "conclude off-field requirements and team planning" for the Lions series.
"Although not an official Wallabies squad, the selection... does reward a number of players who have shown strong form in this year's Super Rugby competition," an Australian Rugby Union release said.
Eleven ACT Brumbies players were invited, a reflection of their good start to the Super Rugby season, with in-form but uncapped back-rower Ben Mowen, number eight Fotu Auelua, scrum-half Nic White and full-back Jesse Mogg among them.
There was no place for George Smith, however, as the 110-cap flanker remains ineligible due to his contract with a Japanese club.
Cooper's omission and the indefinite ban for fighting his team-mates handed to Kurtley Beale last month leaves James O'Connor, Berrick Barnes and uncapped Christian Leali'ifano as the fly-half options.
Cooper fell out with Deans in spectacular fashion last year when he slammed the coach's tactics and what he said was a "toxic" atmosphere in the Wallabies camp.
The 25-year-old Queensland Red said on Thursday he thought his relationship with Deans was now normal and his exclusion "no big issue".
The first 25-man Wallabies squad for the Lions series will be named on May 19 with a further six players added when the Super Rugby season pauses for the international break.
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