I was going a bit of digging and realized that generation 4 games use the same origin game flags as generation 3, and the generation 4 game flags are handled gracefully when on Pokemon in generation 3. So, I tried putting the game origin of HeartGold on an Unown as I transfered it from Gold to Emerald, and then moved the emulated save onto a real cartridge. From there, I was able to transfer it through Pal Park, and the origin flag was maintained. At this point I was super hopeful and finished the transfer up to Home, and the flag was maintained the whole way!
This means that I can maintain the proper region of origin for any Pokemon transferred through Lanette’s Poke Transporter!