"He's still got niggles with his ab," Cooper told Newstalk ZB's Mark Watson. "And [fellow loose forward] Lachie Boshier won't come either [due to appendicitis]. The rest of the boys got through the game pretty well."
The Chiefs will take a squad of 29 to South Africa as cover.
They have received financial support from New Zealand Rugby so All Blacks Brodie Retallick, Nathan Harris, Anton Lienert-Brown and Damian McKenzie can return home after the first match against the Stormers and before the second against the Sharks.
The Chiefs' defeat presented the Jaguares with another pioneering moment after last week's win over the Blues — their maiden victory in New Zealand. The Argentinian side completed a perfect four-game road trip with earlier triumphs over the Brumbies and Rebels.
Cooper offered no excuses.
"They certainly shut us down defensively and their lineout work was good. They're an international team, that's probably what [fans are] not realising. That's the Argentinian side out there. We certainly didn't take them lightly.
"We had a number of opportunities — a disallowed try [for an earlier knock-on] by Charlie [Ngatai] was a big moment [at 23-12], then an attacking scrum when the ball popped out after we thought we had dominance."
Cooper said yellow cards to Michael Allardice and Messam, and a penalty try, also added up.
"The Jaguares player drove downwards and Liam drove with him, so the yellow card-penalty try was a hard call I thought.
"But our discipline has to be better. There were changes in our team but we were good enough. We could've, and should've, won the game."
