I feel for you, I had the same experience with my son every night in the first couple months, wanting to nurse every 30 minutes, and then going back and forth between breasts even when he had already cleaned them out. It was frustrating, and even though he never really seemed content he would eventually fall asleep while sucking. My milk increased as I demand fed throughout the day and night, even though it often took a few days to increase enough to satisfy him at night at the stages when his demand increased. Have you already worked on increasing your milk supply (i.e. demand feeding, drinking lots of water, taking a supplement like Fenugreek, pumping when he's not nursing)? I'm sure you know that once you start supplementing, it just spirals from there. Your body will continue to produce less milk and you will begin supplementing more, and so on.
As for the formula, after we supplemented with formula a tiny bit when we first brought our son home (trouble with breastfeeding), I read more about how the baby's gut works. You will probably never hear this from a doctor, but a baby has an open system until about 6 months of age, which is how immunities are transferred by the mother's milk into her baby's bloodstream - through his "open" gut. So if you use formula, especially when they are closer to newborn, any substances (i.e. milk protein or any additives) in that formula are going to be able to cross over into the baby's bloodstream - hence the development of allergies, eczema, etc. (same thing that can happen in an adult who has a "leaky gut"). After I learned this, we never touched formula again. So if at all possible, hang on until 6 months and then supplement if needed.
If you do end up having to supplement, just know that there are other options out there, including donated breastmilk and homemade formulas (I know doctors would want to hang me for suggesting that) like the Nourishing Traditions formula that people all over the world successfully use with their children. Also, while straight goat milk would be nutritionally deficient to nourish a newborn full-time, if you only used a little to supplement the extra hunger, the breastmilk should still provide all the necessary nutrients (but to provide much nutrition the goat milk would need to be fresh from the farm).