Remember a BOB is only for short-term, temporary, living...you use it long enough to get to a more permanent & secure shelter/home/location. BOB means Bug out Bag meaning you use it long enough to go to somewhere else. It isn't called a "live out of it for a long duration or forever bag."
Depending upon your system & plan, it could be 1 to 7 days tops. I think sometimes people get confused and think they're going to ride out an entire catastrophic event living out a backpack. Heck even in the Military they don't live out of a 95 pound pack for months on end without some form of re-supply, or recovering a cache or collaborating with an indigenous people.
So, depending upon your system/plan you may not even need a cleaning kit for your BOB at all since you're only going to live out of the BOB for a few days or a week at the most until you arrive at your Plan A, B, or C location.
But if your Bug Out Plan calls for a cleaning kit then with all this in mind, go with a Bore Snake and a small bottle of CLP. It is all you need until you arrive at a more permanent/secure shelter/location. You just need to keep the firearm operational and not spotless. A firearm can function quite well with a dirty bore until you arrive at a more permanent/secure location. Then you can break out the deeper cleaning kit.
But if you're concerned about mud or rain then a Bore Snake won't work for the mud but will be fine for rain. For mud you might want to find a cheap rod. You don't want to invest too much in a BOB because what happens if you lose your pack or worse it get's taken/stolen? Which during a potentally hostile catostrophic event it is very likely.