How can I further my dog treat dispensing product?

You may improve your product's credibility by getting endorsements from vets. Consider partnering with local/regional animal welfare groups.

Hello! Congratulations on your effort thus far! Keep at it and you will make things happen.
My name is Humberto Valle, I'm a marketing strategist with almost 10 years experience and currently head the development at www.Unthink.me.

One of the first things I want to share is that Newman's Own charity angle is just that, an angle. Yes, they do good but at the end of the day is a marketing tactic to gain an edge with a possible under-targetted population. Having this approach is not wrong at all, but it is important to realize that marketing it needs to rely on the core value and features of its utility to the right persona who will care about the cherry on top being the charitable donations.

Anyway, my main suggestion for you to answer your question was that you should try a crowdfunding approach, something like kickstarter.com or indiegogo.com. If anything, simply to measure people's interest and to get feedback from real life conversations. Try leveraging email marketing as much as possible to get exposure, get yourself a simple landing page (s) to test different messaging for different people and redirect attention to fund pages as well as your story - what makes yo likeable, believable as the one heading this effort and why you want to help animals and how.

I would look at all possible angles particularly the medication perspective as this may be your big ticket selling point. Many of us pet owners find it hard to keep track of meds particularly when we have more than one furbaby.
Reach out to vets as some previously recommended but you may also want to reach out to a manufacturer like Merck Animal Health. Your product could improve their sales and their products performance if meds are distributed more consistently.
You might also want to reach out to private shelters - rescue groups and animal sactuaries, even zoos. Big Cat Rescue in FL, Liz E's Pekingese Rescue in Arkansas. They could share with you the challenges they have in making sure animals are getting their meds and/or treats.
I'd be happy to discuss further if you'd like.
Best of luck!