Angela here from Mixpanel Support!
You can definitely customize the phone number or Sender ID when sending SMS from Mixpanel, but you'll need to use your own Twilio account instead of Mixpanel's. You can add your Twilio account info in the Project settings, Messages tab:
You can then email email@example.com and ask for 100% of SMS messages to be sent through your Twilio with the customized number.
Hope this was useful!