The only way I can imagine achieving this is by mounting a player onto a horse with a customized model, so it displays the desired entity.

Unfortunately, I don't think there is any other way or workaround. It cannot be done without client-side render mods. However, even with client-side mods, you can still stay on the Bukkit/Spigot/Paper servers and build communication between mods and plugins over the Minecraft Protocol.