Halo 2 is quite special when it comes to co op even on the OG Campaign. I've noticed myself that those lag outs happen in my mod when a player gets into a Scorpion (vanilla campaign bug) thats relatively rare and another instance that pertains to the Spartan AI, due to how their animation graph is set up/how I have it parented to the OG spartan graph. One or more of their animations is causing a de-sync making that pesky lag out bug.

Therefore I suggest only playing Coagulation and Waterworks for now (Coag has lasted the longest in co op out of those two) as Sanctuary and Warlock will have you lagged out in a few minutes or less due to the bug I mentioned.
I have been working on said issue by deleting any animations that are not imperative to the overall design of the AI basically trying to narrow down and minimizing the issue as much as possible until it can be identified what animation is causing it and be deleted/fixed.

(Edit: the above issue of co op de syncs has been completely fixed)


Also not much you can quite do about the lag as campaign operates off of p2p connection so location is gonna factor in quite alot with performance, even on fast ethernet connections. It is possible as i've played with friends who have had no lag and some who have even with Ethernet.