IIS 8.5 on Server 2012 R2 stops responding after a time

I recently updated a server from 2008 R2 to 2012 R2 that hosts a few MVC sites and a Web API. After deployment, everything seems to work fine for a few days before the web api seems to fail. It returns a 404 for all requests to the API from the failure forward. The parent MVC site seems to continue working fine.

A few things:

• The web api is hosted as a web application inside a parent website within iis.
• This is a 4.6.2 framework site and api.
• I would prefer it to be its own site but I don't control this.
• It seems to stop at roughly the same time when it occurs - around 2:35 AM
• The only route defined by the web api is a GET

I've checked the event log as well as IIS logs. The event log doesn't reflect anything during these times and the IIS logs just show a 404 response. Resetting IIS/AppPools/etc... don't fix the api nor does restarting the machine. In fact, the only thing that seems to fix it temporarily is a VS publish over top of site.

I suspect something to do with MSDeploy but have nothing concrete. Does anyone have any ideas on where to look or what to look for? I feel it must be something to do with the server configuration as we've never seen this problem prior.