Is there any API for my mod to communicate with so that I could load a custom HD texture onto my dummy Player Entity? Thank you!

We support loading player skins in a similar way to the Mojang API. You can actively add a GameProfile for a given player in the spigot plugin, like you said in this video https://www.youtube.com/watch?v=vswQvReadrc.
Unlike the Mojang API, the base64 parsed address in the texture attribute can point to any address of legal png image, not just Mojang's addresses, so you can add the address to the HD skin here.