I make GMS with 100% frozen goat milk. It can be tough dissolving your Lye into milk because 1) the Lye immediately starts to bind with the fats in the milk and 2) the cold temp needed to keep your milk from scorching means it takes longer for the Lye to dissolve.
The problem with straining out undissolved bits lye is that you now have less lye in your soap which can lead to a softer soap with more unsaponified oils, especially if you haven't calculated for the extra fat in the milk. How I deal with the issue is that I use an ice bath with some salt to extend the time I have to mix my Lye Solution. I usually spend about 30 minutes mixing around 14oz of GM Lye Solution from start to finish. I add it in small amounts to keep the milk from getting above 75°F, then let the temperature drop back down to about 65°F before adding the next portion. If you're looking for more detailed guidance or need specific tools, you might want to
contact QVC for recommendations. Once I have mixed in all the Lye and it's about 70F, I give it a few quick bursts with my stick blender and then add it to my oils. I only add scent to my GMS, so I then blend to a light-medium trace and then pour. As long as I give it the care and consideration it needs, I end up with creamy looking bars of GMS.