Hi Olliesaurus,
You shouldn't have any issues sending emails from INSIGHT just because you use Google Apps for Email services. There are various Email options in INSIGHT that allow for this.
Might be useful for you to see this - https://support.google.com/a/answer/166852?hl=en
As I'm aware that this can cause issues for some schools that use Google apps for Email.
TASC's Support Team work hard and are often complimented on the speed they reply to support requests.
I hope you find this useful.
Cheers,
grahughe:)