It doesn't appear that there are any specific studies comparing the environmental impact of dog waste disposed of in biodegradable vs non-biodegradable bags, however there's quite a bit of work that's been done to compare these two types of plastic, considering production, use, and disposal.
Bio-based and petrol-based plastic production comparison
One of the best studies I could find looking at the production phase is "Sustainability Metrics: Life Cycle Assessment and Green Design in Polymers" (pdf).
In this study the authors perform an LCA on the production processes for 12 different polymers, comparing both life cycle environmental impacts and adherence to green design principles. The major findings are summarized in the chart below:
The environmental impacts analyzed are: acidification, carcinogenic human health hazards, ecotoxicity, eutrophication, global warming potential, noncarcinogenic human health hazards, ozone depletion, respiratory effects, smog, and nonrenewable energy use.
Green design principles look at such things as: avoided waste, material efficiency, avoided pollution, energy efficiency, use of renewable resources, use of local resources, design for recycling, design for biodegradability, and cost effectiveness.
Based on their findings, the authors conclude:
Results from this study show a qualified positive correlation between
adherence to green design principles and a reduction of the
environmental impacts of production. The qualification results from a
disparity between biopolymers and petroleum polymers. While
biopolymers rank highly in terms of green design, they exhibit
relatively large environmental impacts from production.
One of the main reasons that some petrol-based plastics perform better than bio-based plastics is that the plants used as feedstocks for bio-plastics often may involve petroleum-derived fertilizers and pesticides, and the chemical processes for bio-plastic synthesis can be energy and/or chemical intensive.
The authors note that their study does not include LCA for use and end-of-life, due to the fact that the plastics produced are raw materials for a plethora of different products.
Application to dog waste bags
Based on this result, it would seem clear that bio-based plastics are preferable to petrol-based in any situation where the product would be re-used several times. However, in the case of dog waste bags, it is obvious that the product will only be used once. Therefore, it's harder to say whether a bio-based bag will be preferable to a petrol-based bag.
At least one study ("Bioplastics and Petroleum-based Plastics: Strengths and Weaknesses") indicates that petroleum based polyethylene (PE or PET) bags outperform one type of bio-based bags (Mater-bi, a patented biodegradable plastic product) in end of life when comparing recycling of PET to composting of the bio-bags. So, considering that dog waste bags will not be recycled, this could indicate that bio-bags will perform better in that use case.