Unfortunately, I can’t say much. I never took out private loans, and you’re right: They almost universally cannot be consolidated via the federal government. My best suggestion is that you contact them and explain you’re experiencing financial hardship (even if it isn’t true), and ask about paperwork for an income based repayment plan. They may or may not have one. Also ask about whatever incentives or forgiveness programs they may offer (like if you work in a school affected by poverty, or other types of public service).
If anyone out there has some solid advice to add, please do so.