Configure Twilio SMS Forwarding

My booking system already sends confirmation and reminder texts through Twilio, but when a customer replies they receive the default “number not configured” notice. I need the Twilio number properly configured so every incoming SMS is automatically forwarded to a single email inbox.

Here is what I know so far:
• Forwarding choice: I only want the replies delivered by email, not handled inside Twilio.
• Email platform: it is an “other” provider (not Gmail or Outlook). I am unsure whether the service exposes an API.

What I need from you:
1. Set up an inbound SMS webhook for the existing Twilio number.
2. Build a lightweight relay—Twilio Function, serverless script, or any secure method you prefer—that converts each incoming message (sender, body, timestamp) into an email.
3. Point the email to the address I will supply, preserving the original phone number in the subject line for easy filtering.
4. Provide a brief walkthrough so I can change the destination address later if needed.

Acceptance criteria:
• When I text the number from my own phone, the message appears in my inbox within a few seconds.
• The default “configure your number” auto-reply disappears.
• Your code or settings are documented in a short read-me.

Please bring any preferred tools—Twilio Functions, SendGrid, AWS SES, Node.js, Python—whatever achieves clean, reliable forwarding without adding monthly costs beyond Twilio’s standard fees.