This requirement is clearly the bottleneck that would cause Scouts to back up in our troop as well, when you have a full patrol of new boys. We, like Twocubdad, also have various months where the outings are not organized in a way to allow for patrol cooking.
One can imagine a variety of methods to deal with this. A couple that we've done are 1) to have six-meal campouts, where two Scouts can fulfill the requirement, and 2) to throw in an extra patrol campout or two, just for the New Scout Patrol.
An obvious option is to:
3) Start assigning a head cook on campouts as early as pos