Your insight suggests an even better solution: the spacewalking astronaut could open the internal hatch to the docked Lunar Module so its cabin air can instantly partially repressurize the Command Module
I mean we're just guessing and spitballing here, and all of this assumes that whatever incapacitated the CMP is fixable by the other astronaut. But it does look like there may be a decent chance the CMP could survive this. Of course, he might have internal injuries or they might not be able to restart his heart. But seems like this procedure would be worth a try.