IANAL/IANYL. That said, the GPL's obligations (to provide source, to license it under the GPL, etc.) cut in when you distribute (or convey, in the language of GPLv3) copies of GPL software, whether modified or not. Making, modifying, and running copies for internal use does not count as distribution.
So: as long as you are hosting Mosaico rather than distributing copies of it, GPL obligations will not cut in. That was the whole reason for the development of the Affero GPL (if mosaico was AGPL-licensed, this argument would not apply). You can run mosaico on your servers, you can modify mosaico to better run as part of your SaaS offering, and it can talk to your proprietary software any way you want, without triggering any GPL obligations.
Note that all this development and deployment must be in-house, that is development must be by your company's employees and officers, and deployment must be to equipment that your company owns or rents. If you start using external contractors, and giving them copies of your source to work with, that may well count as distribution. If you deploy to some other organisation's servers, that likely counts as distribution.