WordPress generated emails not sending

Hi, I site, where users submit applications via gravity forms. Once this happens various notifcations are sent using gravity forms and post status notifier.

The problem is the after about 200 of these are sent each day, the notifications stop sending, which is obviously a huge problem. I know gravity forms isn’t to blame because I have a DEV site on a different server which is working fine. So obviously it is a shared server issue, I have checked with the hosting company to see if the site is being blacklsted, but they are less then useless.

I understand that the site uses wordpress@sitename to generate the notifications and potentially the server could only be allowing 200 emails a day to be sent. I asked the host to change this to 1000 but there are no email addresses associated with the hosting. Not sure what to do here, any advice would be great.I have also tried using a gmail SMTP but this does not work. Any help please.