Need help with best way to dynamically route [Diagram included]

Please see the following diagram
https://imgur.com/gallery/p0QC6yD

This is our current configuration for our connectivity to a service provider. My org are the sites, their org owns the routers and datacenters. The datacenters must be inter-connected since the IPs that we route to are the same no matter which route we use. I do know they use BGP on their end to dynamically route. Also our internal IPs are NAT'd / NAT overloaded on their routers.

Right now, If Datacenter 1 or Router 1 goes down HSRP fails over to Router 2 and all traffic automatically goes to Datacenter 2. If Datacenter 2 goes down or Router 3 goes down, I have to manually change static routes at Site 3 to tell it to route through Site 1. The same if Site 1 were to go down, I would have to manually change Site 2 static route to point to Site 3.

Is there a way I can use BGP to neighbor with the service provider owned routers without advertising my own internal routes? If they inject the BGP routes into OSPF and I distribute those, is there a way to avoid Site 2 doing ECMP? I admit I'm a jack of all trades and I don't understand BGP well but what do you think the best way to dynamically route this would be?