All the recipient's carrier knows is they see a spam or phishing message originating from Twilio, and (profess to) have a zero tolerance for spam/phishing. They have no idea about context of the "forward to consenting employee" use case or any way of validating it.
If there is any possibility that those messages could be interpreted as spam, that doesn't seem to be a valid use case for Twilio (any more). You could forward the message over some channel other than SMS to the employee (like email, or a custom solution).
If there is any possibility that those messages could be interpreted as spam, that doesn't seem to be a valid use case for Twilio (any more). You could forward the message over some channel other than SMS to the employee (like email, or a custom solution).