==> SOLVED <==

I've been able to recover the correct private key from the backup I took before deploying the new installation and loaded it as key to use.

Would be nice to have an official solution method from Odoo tho...

Yes, you're on the right track. Since your previous Odoo CE instance is gone but the EDI proxy still holds a registration with your VAT and old key pair, the current installation can't fully sync (especially for status updates and incoming invoices).

✅ Here's how to reset/renew the connection properly:

1. Regenerate the key pair in Odoo:
   • Go to Invoicing > Configuration > Electronic Invoicing > EDI Formats.
   • Open the FatturaPA (Italian e-invoicing) format.
   • Click "Reset Connection" or "Renew Keys" (wording may vary).
   • This regenerates a new public/private key pair and re-registers with the EDI Proxy using your current Odoo instance.
2. Ensure Odoo has internet access and the correct company VAT:
   • Make sure the VAT number in Settings > Companies > Your Company is correct.
   • Odoo uses this to register with the proxy and associate with SDI.
3. Wait for the registration to complete:
   • Check logs or status updates. Once confirmed, incoming SDI updates and supplier invoices should start coming in again.
4. (Optional) Contact Odoo Support or your proxy provider:
   • If it's stuck (e.g., key mismatch persists), they can manually clear the old key pair or re-link your VAT to the new one.

Let me know which Odoo version you're on if you want the exact UI steps.