You cannot re-sync a DualShock 3 / Sixaxis without a USB cable. The Bluetooth pairing handshake requires a wired data connection to exchange keys.
Sony designed the DS3 to pair only when physically tethered — a security measure to prevent unauthorized controllers in tournaments / shared spaces. The PS4/PS5 DualSense/DualShock 4 support wireless pairing via Settings → Devices → Bluetooth Devices.