Goat's milk is not a better option than cow's milk in terms of nutrition. Just like you don't give an infant cow's milk (regardless of lactose intolerance), you can't just start giving it goat's milk because it lacks nutrition.
Using goat’s milk before 6 months or regular use between 6 and 12
months is not recommended. Goat’s milk is no more appropriate to give
baby than cow’s milk. If you need to supplement and breastmilk is not
available, formulas are a more nutritionally complete product. There
are several comparisons of goat vs. cow vs. human milk in the links
Goat’s milk is high in sodium (like cow’s milk) and is
very high in chloride and potassium, which makes the renal solute load
too high for babies. This can cause gastrointestinal bleeding and can
result in anemia and poor growth (these problems are usually
undetected until months later). Goat milk is also deficient in folic
acid, which can lead to megaloblastic anemia. Also, infants who are
allergic to cow’s milk protein are often allergic to goat’s milk too.
While it’s true that whole goats milk (and whole cow’s milk) was
commonly used prior to the advent of infant formulas it is also true
that the infant mortality and morbidity rate during the times of such
substitutions was very high. source
You can also read about the dangers of goat milk from the American Academy of Pediatrics (AAP):
Goat's milk contains 50 mg of sodium and 3.56 g of protein per 100 mL,
approximately 3 times that in human milk (17 mg and 1.03 g per 100 mL,
respectively).6 The estimated requirements of sodium and protein for
infants <6 months old are 100 to 200 mg/day and 9 to 11 g/day,
respectively.7 The infant described here was receiving ∼500 mg/day of
sodium and 30 g/day of protein, with a total intake of 32 oz of goat's
milk per day. The immature kidneys in very young infants have
difficulty handling the byproducts of foods with a high renal solute
load.8 Sodium excretion capacity matures more slowly than glomerular
filtration rate and does not attain full capacity until the second
year of life.9 Therefore, infants fed fresh goat's milk are at
substantive risk for hypernatremia and azotemia, particularly in the
face of dehydration (as in the case described here), which may in turn
result in major central nervous system pathology, including diffuse
encephalopathy, intraparenchymal hemorrhage, or thromboses10 as
manifested in our patient.
This article from the Canadian Paediatric Society describes an infant with severe anemia and a murmur due to a diet of exclusive goat’s milk:
Goat’s milk is known to be deficient in vitamin D, vitamin B12, iron
and especially folate. Infants younger than six months of age need 65
μg/day of folate (the recommended daily allowance increases with age).
Goat’s milk contains 6 μg/L of folate (breast milk and cow’s milk
contain approximately 45 μg/L to 50 μg/L). The infant’s serum folate
was less than 1.4 nmol/L (normal 7 nmol/L to 39.7 nmol/L), her serum
vitamin B12 was 141 pmol/L (normal 200 pmol/L to 540 pmol/L) and her
serum iron level was also low. The diagnosis of macrocytic anemia
secondary to a combined folate and iron deficiency was made.
Young children should only be given pasteurized milk; please read about myths around raw and pasteurized milk. Raw milk isn't sterile (like breastmilk) and may carry serious diseases (source 1, source 2, source 3). People do die from it. Having said that, my relatives own a dairy farm and they all swear by raw milk - none of them have ever been sick. I also know a 90-year old man who has been smoking since he was 11, and he doesn't have lung cancer.
Since breastfeeding is no longer an option, you are going to have to try formula. In an effort to promote breastfeeding, there has been a lot of negative propaganda around formula. However, formula closest represents breastmilk. It isn't equal by any means, but a lot of work has gone into making it nutritionally similar to breastmilk.
I can understand your concern about sugar in formula, however, even breastmilk is a lot of sugar - it just happens to be provided by lactose. Sugar also provides the carbohydrate source that babies need:
For infants with lactose intolerance, formulas that contain no lactose
can be used. Lactofree is an example of a milk-based formula that
contains corn-syrup solids rather than lactose as its carbohydrate
calorie source. Many soy-protein formulas also do not contain lactose
and are suitable for lactose intolerant infants. In addition to
corn-syrup solids, other examples of carbohydrates contained in
lactose-free formulas include sucrose (table sugar), tapioca starch,
modified cornstarch, and glucose polymers (short chains of glucose
I found one product in US that doesn't use cane sugar or corn syrup solids: it uses brown rice syrup as a carb source instead. There are a lot of options in Europe (Aptamil pepti) due to a ban of sucrose in their infant formulas, but I am not sure if you can arrange for that.
I hope this helps.