In the time it takes to figure out a solution and implement it you could do 5 runs to the store and stock up on dog food for a good long while...

Do you really want to break your back/crash/look like a fool while wrestling with a monstruous sack of dog food? I'd just buy the biggest sack that fits into my backpack/messenger bag. It doesn't have to fit in completely, it can stick out the top.

Although I have to say I like the stem+bullhorn rack idea. But you'd need a helluva long stem and bullhorn to get a 35lbs sack to fit on.