Need a Bulk Email / Email Server / Inboxing Expert

I am not sure this is even possible so please only bid after reading every word and feeling 100% confident this can be done. We are an executive search and staffing firm using a shared email server (shared with other search firms who use PC Recruiter- a well known SaaS solution for the recruiting industry) for notifying job seekers about our open positions. I have a feeling, based on the diminishing responses we've gotten over the course of the last couple years, that some ISPs might be blocking the emails from going through. I would like to set up an email server that will get these emails inboxed but still use our cloud-based applicant tracking software (ATS) that houses our database of candidates, helps prevents us from sending duplicate emails to the same email address, manages our unsubscribe list, and allows us to filter by zip code radius. The ATS allows us to choose an outgoing SMTP server, user name and pword. I'm hoping that it is possible to set up our own email server where we can provide the appropriate SMTP server address / user name / pword and whatever else is needed to guarantee the emails get inboxed and need an expert consultant who can help us get this set up. I've been told that since we don't own the server from where the emails are originating that this cannot be done. But I'm hoping someone knows a way to do this. I am using Cox cable internet in my home office as an internet service provider. We need something that requires little to no maintenance. That will just sit there and process all the emails we need to send and have those emails get through spam filters.