The old Alpha Strike Companion has a section for campaign rules that are at their heart a version of the Chaos Campaign rules tailored to Alpha Strike. The biggest thing to get from this is how to determine if a pilot is wounded or killed if a unit is damaged/destroyed, as Alpha Strike doesn't really separate pilot from unit like classic does. There's a nice table for it in there.
That being said, there's nothing in the Chaos Campaign rules in Campaign Operations that's inherently incompatible. Objective and bonus Warchest Points don't really need to be changed between using one system or the other, but some of the objectives that require "X number of turns" might need to be tweaked to account for Alpha Strike's faster play. The scenarios in the ilClan sourcebook give some good examples of that.
Support points conversion shouldn't be very different either. Outside of determining if a unit is only using introductory tech if you aren't familiar with it from Classic, but the Master Unit List can help you there. The other thing would be ammo costs, you could ignore it since Alpha Strike doesn't track ammo, or make units that don't have the ENE special pay it between tracks, or pay it if you want to use special munitions.
