Lysine is not an antiviral. Lysine is an essential amino acid.
EBV (even without rapa) is a huge and complicated issue, and the linked to paper in this X-thread is about future research, and nothing currently actionable.
Alternative medicine communities have discussed the use of BHT (more commonly used as a food preservative) for decades as a potential treatment for herpesviruses.
That said, I know of no no large-scale FDA-approved clinical trials confirming its efficacy for EBV or infectious mononucleosis in humans.