The 'caregivers' in this case are professional veterinarians who think the cure can be worse than the disease. The caregivers allege that the chimps are being cared for at a standard that is as high as the one they'd get at the Federal facility, without needing to move them, which causes stress and jeopardizes their well-being as well.

Part of the problem here is that the law seems to say "they will be moved to the Federal sanctuary," not "they will be cared in [list of appropriate ways]". To fully comply with the law, you need to move them even if they're sick, have social bonds in their current place, or are perfectly happy.

Disclaimer: I've visited one of the non-federal facilities and I've seen retired chimps that looked content. I have zero experience in animal welfare.