Can you take naproxen 500mg then take either tylenol (acetaminophen) or aspirin for break through pain. Are this meds ok to take with elevated liver functions?

Safer alternatives. In light of a history of liver dysfunction, acetominophen (tylenol) should be limited. Since you are already looking for something else to help beyond naproxen, it is time to ask your medical professional for better and safer options. Some alternative treatments or "stronger" medications will actually be safer over time with less chance for liver or kidney toxicity.
Yes and no. Aspirin (except for low dose, cardioprotective doses) should not be taken with naproxen. They share similar mechanisms and their toxicities are additive. Tylenol (acetaminophen) (not exceeding 4, 000 mg/24 hours) is okay. The combination if prolonged can affect kidney function. An overdose of tylenol (acetaminophen) causes liver failure. For severe pain, seek medical attn for reversible causes and/or stronger relief.