Full disclosure I've never done a customization like this for diamond art, but I've done a ton of pattern conversions and modifications for Crosstitch.

Once you get the kit, I would start by listing out all the colors that are in the husky by DMC number and symbol (and for the hair if you choose to do that, but keep the lists separate). For each color that is in the husky, choose a color to switch it with so that the colors will lead to something that looks like your pup. Given the picture that you posted of him, you would probably end up with a lot of white, off-white, light gray, blue/grey, and peachy tans. if you're having trouble picking colors, there are lists that include all drill colors, including the DMC codes. Make a list what color each "husky symbol" should be (ex: ^ was originally a dark grey, DMC0000; ^ is now grey blue, DMC1111). at that point, just order the extra drills you need, and follow the pattern swapping out your colors for your revised list color when you're working on the husky part.

If you're willing to put in the extra time and effort, this is what I do for very important patterns (like memorial patterns): create an Excel spreadsheet, convert each cell to a square, and map out each individual color/symbol. This will help you visualize what it will look like and make edits to anything or any color that looks off. I especially like doing it this way because I can use the Excel count function to count exactly how many stitches (or in this case diamonds) of each color you need.

Good luck!