You can try out Halo 2 Livelessness mod. Basically it restores your LAN play at the cost of the cartographer MOD.
https://www.halo2.online/threads/xlln-xlivelessness-lan-enabled.2769/
I got the LAN enabled using the mod from this thread, but the cartographer doesn't seem to work.
- Download the Xlivelessness from the links from the above thread.
- Take a backup of xlive.dll from your halo2 directory.
- Extract the contents of the download to your Halo 2 directory.
- After you startup the game, use the network tab to create/join games.
You'll need to do the same for the other PC on your Local network and find out to see the games you have started.
You can also use an external Frame limiter to remove stuttering, and get improve mouse response by using MouseThingy.
Also ensure that you have selected appropriate internet adapter in settings>network.
You'll not be able to play online multiplayer games and will need to move the original xlive.dll to revert to project cartographer.
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Let me know if it works for you.