Cheika had requested the Wallabies trio be rested as the match against the Japanese team in Canberra falls just six days before the opening test of Australia's series against Six Nations champions Ireland.
The Brumbies, however, are engaged in a battle to arrest a worrying decline in crowd numbers and were reluctant to rest three of their best players for an afternoon match they hope will attract 15,000 fans.
The side can also still make the playoffs as they sit 11 points behind the Australian Conference-leading NSW Waratahs with four matches remaining, meaning a bonus-point win on Sunday would help them stay in the hunt.
Openside flanker Pocock, and props Sio and Alaalatoa will all start against the Sunwolves along with fullback Tom Banks, centre Tevita Kuridani and scrumhalf Joe Powell, who were also named in Cheika's squad for the June series.
A seventh player in the Australia squad, uncapped hooker Folau Fainga'a, is suspended after being sent off for a headbutt in the victory over South Africa's Bulls in Pretoria last week and will be replaced in the front row by Connal McInerney.
"We don't play 15 or 20 home games. We play eight, and people pay to see Pocock, Sio and Alaalatoa, they want to see them week in and week out," coach Dan McKellar told the Canberra Times.
"We just felt the scheduling and the timing of (the request) was a little bit off, and at this point in time we need to do what's best for the Brumbies.
"We both understand each other's positions, in this instance we can't come to an agreement. It's far from ideal ... and we need to make sure moving forward this doesn't happen again."
Brumbies officials have warned that the twice Super Rugby champions might "cease to exist" if crowd numbers did not improve but Cheika had suggested fans in the Australian capital would turn out even if the trio were not playing.
"They're a good crowd down there in Canberra," the coach said this week.
"They've always been very supportive of their team and I'm sure they'll be out there in force this weekend, regardless of whether those three lads are playing or not."
The Sunwolves will be resting their Japan internationals for the trip to Australia ahead of June tests against Italy and Georgia.
(Reporting by Nick Mulvenney; Editing by John O'Brien)
