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Another possibility comes to mind: use a single-byte escape code instead (ASCII's Data Link Escape code 16 seems appropriate), and add a modem command that sends that literal byte and goes back to data mode.

So, for example, whenever you need to transmit `␐`, you'd instead transmit `␐ATY␍` (for a hypothetical command `ATY` that sends a literal ␐ and switches back to online mode) and then continue transmitting as normal.