Exped makes some good 3 and 4 season tents that are exoskeleton tents, which means that the rain fly is independent from the main portion and can be pitched solo in the summer months to cut down on some serious weight. I have the Venus 3 and it's a solid tent.

Suggest you look at the Eureka Zeus family of pack tents.

They're an ex-skeleton single wall so very light but with enough room to sit up which can be very important on inclimate days.  

Unfortunately I think you will find a real problem in product offering for a family pack tent.  Most of the designs are either for a maximum of two or weight prohibitive around group camping where the load bearing is shared amongst all the hikers (like Military tents).  

If you are lucky you may find a Zeus III which should fit all your family in a nice light weight but most in that size, 7x7, are going to be over 7lbs, like the Eureka 4.  

An approach would be buy a Zues II, 3.5 lbs and then a second Zues I, you carry the II and the wife the I.  When you setup you can either put three of you, the little one in the II and older child in the I nose to nose at first till they get use to being in their own tent or Mom with the two little ones in the II and dad in the I.  

For a BOB, I don' recommend tarps.  Tarps are actually my choice for a GHB which is a most likely one night or three at most.  On a bob where you may be looking at God knows how long, a free standing tent is the only way to go.  The longer one stays out there, the more important things like keeping bugs off you, keeping dry, and keeping warm becomes.  Free standing is also important in that it gives you the option of setting up on rock, asphalt, or concrete where driving stakes is not practical and there aren't any trees.  

Also for a BOB weight is everything.  The more weight you lose on carrying things that could be lighter the more important things like food and water has to be left behind.  It was Lord Wellington during the Napoleonic War in Spain that said he can always follow an Army by how much stuff the soldiers discard from their packs.  We will be no different in some sort of SHTF scenario as we start out with what we normally carry then as the days wear on start chucking gear to save weight.  The more food one has, the better because food has a natural attrition, eating, which reduces weight and it keeps our strength up so we can carry more longer.

I encourage everyone into BOBs to use their gear but will take it a step farther, use it in the worst conditions, because a fair weather BOB in a crisis is about as useful as a fair weather friend.

Tj

Hoofing it with two kids under four is going to be a bigger problem than finding a tent. I used to take my grandson hiking with me on a 3.5 mile local loop trail and he could never finish it at that age. I always ended up carrying him for the last mile or so. (It was hilly too)

I suggest you accept the reality that you're not going to get very far on foot with kids that age and stick with BOV gear for the time being.  Or, if you only have to go a short distance on foot, carry the kids in and you can go back yourself to hump the gear in on a second trip.