I don't know that any hard data exists on this because, frankly, it seems kind of silly. Plenty of companies have been incredibly successful without any sort of character as part of their branding (Uber, Twitter, Facebook, Google, Apple - to name just a few) and there are plenty of "characters" in the graveyards of internet startups. Sure, some successful ones may stand out to you, but is it because the characters sold the product, or is it because the product itself was so good that you got a lot of exposure to the characters?

It doesn't seem like design/marketing is your area of expertise, yet you are trying to push someone who is more experienced in that area to follow your vision, which you don't really have any good validation for (hence the reason you are asking here) - that doesn't seem like very good leadership to me.

You are still building a product - is it worth expending this much time, energy, and conflict with your designer over something that most of us think is pretty trivial? Is your product in a good enough state that you have the luxury of fighting over something like this?