Probably a large ask of your doggo, but have you tried a slightly long-term exposure? I know longer term exposures will smooth out the ripples and provide a better glass-like reflection, but I’m unsure how long that would be, and whether or not your dog would willingly sit there for that long.
Another, more Photoshop heavy solution is to take multiple photos in burst mode (say, 5x 1/800) and then load them all in a single Photoshop file, align them, and do median blending. This should smooth out the ripples as well I think, although I'd only apply this to the bottom to ensure you don't get artifacts on doggo.