... but surely it would be ok, if ...This is a legal question, as well as a technical one. You should probably consult your lawyer. But first, why not take the easy path? Just sign up for the free trial at Constant Contact and see what they can do for you. They've been doing this for quite a few years, I've used them extensively, and they are very good at what they do.
So would it best to use, or is it possible to use SendGrid or something similar so that I can send it from a different email domain?
Do third party email systems allow you to do that?