The OP asked about a Bug-Out scenario so I'm going to assume that for whatever reason, the plan to leave home base was implemented and they are going somewhere in specific. This is where the pretenders get separated from the planners.
You must have someplace to go where there are cached supplies because you won't survive off the land as a refugee shooting squirrels and rabbits in the "wood" somewhere near a "forest" unless it's somewhere in the very deep woods where you already know how to trap and forage. For this purpose, a refugee is someone who isn't at home base and is transiting to get to one using what's in their car/boat (soon to be disabled) or their back. So I'm going to interpret your question to mean "should I take two firearms while travelling from Home Base to Backup Base and should one of them be a .22LR" Well, if you are doing this solo, then you're already in serious trouble if only because you can't post security so pick one battle rifle and have 5-mags for it. If not, then for groups of 3 or more, one of thes guns should be a Ruger 10/22 (or similar) and the rest should be battle rifles using a uniform magazine. A shotgun & ammo is way too heavy for transiting and you should be carrying weight in water/food and avoiding making noise at all costs which means no hunting for birds along the way. Your battle rifles and cross-comm gear while transiting are only for breaking contact with a force, not anything else.
If you bug-out and don't know where you are going, have a plan to get there, have pre-thought out supplies & equipment when you get there, or you aren't experienced enough to survive Indian style in the deep woods, then you aren't bugging-out, you are becoming a refuge and you might as well head directly to the nearest FEMA camp. Thus, my thinking that you bring the .22LR and the battle rifles if in a group of 3 or more but only for the weight savings it will bring that can go to food/water in your packs.
When thinking about this stuff, I strongly suggest you break-down an AR-15 into its upper/lower and stash it inside your go bag along with the rest of your gear (this is actually how I store it) and take a hike. In a true SHTF scenario, you'd be carrying it assembled at some point but when you first bug out, it's just in the pack ready to go with you. The space that gets freed-up will be used to fill your 2nd two-liter camelback because sourcing water is way to difficult/dangerous to make that effort more than you need to. Take the hike - don't eat - pretend you had to leave just then. Where will you get water/food, which way do you go, what are the threats you need to think about, how will you hook-up with your team if there is just texting for 24-hrs, etc? All our plans fall apart once we run random tests. I know. I've been doing it a long time.