Kristen, if you can build something and attach the dog house, leaving that as your nesting area, you might could do it. The general rule is 4 s.f per bird in the coop and 10 sf per bird in the run. Granted, yours will free range while you are home, but what if you take a trip or are gone all day?
Does the dog house have a hinged roof for access? Or will you have to cut a back door or something to get the eggs?
I have one right now that is holding two birds while I finish another coop, and it will become my quarantine shack, but I would never put more than two or three birds in there- way too cramped for more than that.
Are you good at building? Do you have help? Do you have good tools? I am 0 for 3, and so I have to confess I have struggled mightily and shed many a tear trying to build a 4 x 8 tall enough for me to walk into for my Orps. Plus it has almost cost as much as the 6 x8 Kwik Shed from Lowe's and certainly has taken weeks and weeks of my time. I think I will be OK building small, grow out coops and shelters, but even with the Grip tool that thrilled us all a few posts back, I am not sure I would ever tackle something so big again.