Hi Brandon,
Currently, ESG configuration is not synchronised between different NSX Managers; there is no Universal ESG (yet). But that doesn't have to be the case, as you can use Local Egress to create a setup where you have a dedicated running ESG per site and be active for that site. You'll need to use dynamic routing upstream to your provider and use priorities to facilitate the failover between sites and route all traffic through the active site. Not ideal though.
Personally I would do the failover on a higher level, like using DNS.
More information on the local egress: https://networkinferno.net/ingress-optimisation-with-nsx-for-vsphere