Can you explain what the problem you are encountering with the mDP or Thunderbolt to HDMI adapter/s you are using here.
As far as adding anything internally, I don’t see any way that’s just the same as the adapters output. This series doesn’t have spare PCI lines to even connect another GPU board internally. There was a Thunderbolt connected external chassis which could then support a standard PCIe GPU board which could offer a HDMI port. But that’s quite expensive solution and I don’t see it offering anything here. These external GPU setups where intended for rendering videos, not really for the external connection.