I did door to door census work in Detroit when I was 18. There were a lot of stray or neglected pit mixes in my life for a few weeks. Dog spray barely works for dogs. It has a range of about 2 to 4 yards, and the size of the tube means it's hard to aim and you'll miss at first, using at least half the payload in the process. The ingredients in dog spray and bear spray are pretty similar, but I feel like dog spray cans don't have the pressure necessary to prevent you from wearing a lot of the product afterwards. That sucks for a shift, but really sucks on a Backcountry trip.
A lot of folks who carry dog spray do it for liability reasons, in that it's a lot easier to explain deploying a standardized product than giving someone's ankle biter a kick in a lawsuit.
Bear spray works pretty well for everything. I've worked and taught bear safety around Glacier, Yellowstone, the North Slope, and Uqtiagvik. I've witnessed black bears and grizzlies get maced with satisfactory results, and worked with a lot of dudes who have seasoned polar bears.
I often don't carry mace in black bear country unless I'm in a park where bears are accustomed to people. I live in Idaho now and all the black bears are rightfully scared of us.
Long story short, don't compromise. If the extra five ounces for bear mace is too heavy, why carry the initial five ounces of useless dog spray?