Interesting how you are sending SMS with different caller-ID based on contacts, but I am curious - without a VOIP dialer, can you place voice calls from your native dialer with different caller ID ?
Please, please email (or trouble ticket) twilio and ask them to add an "email" verb to twiml ...
That is one of the big missing links in all of this - you should be able to fire off email alerts/messages in your twiml workflow without any third-party[1] integration or other accounts/logins, etc.
[1] I would still consider sendgrid third party since I need a different login, account, etc.