You've gotten some good comments. Thanks so much Dom&O for the EO28 Splash info. I had never heard of it. There are more and more options all the time.
Jen, if your baby reacts to cow's milk, you might not want to give him cottage cheese or yogurt, as you suggested. The milk proteins --- the part of milk that babies are allergic to --- are only partially pre-digested in Enfamil Gentlease. If your son tolerated this, then he might be OK with predigested proteins in cheeses and yogurts (the bacteria partially digest the proteins), but likely not. It IS likely that he's reacting to the goat's milk. His reactions may have evolved into a reflux reaction, which is most uncomfortable when lying down. Eating or drinking more of anything, often helps wash the acid out of the throat. This is why he may be acting hungry. Even if that food causes problems later, it may bring immediate relief.
I understand he still wants to fall asleep the way that babies are meant to sleep, sucking on nourishing liquid. Going back to the formula he did well on is an option I'd lean toward in some cases but maybe here you'll want to wait. Firstly because you said he doesn't want it now anyway. Secondly, I think you'll really want to pin down whether his fussiness is related to consuming milk proteins or not. I think your best bet is to go for zero cow's, goat's, or other milk proteins for a little while and see if his symptoms improve. If not, then something different is going on and you'll want to write back here for some more brainstorming. Read ingredients in everything he's given and avoid whey, casein, cream, butter, cheese, etc.
The EO28 mentioned sounds good. There are some other medical foods that are options as well. There are also amino acid formulas that have no traces of allergenic milk proteins: Nutrimigen AA and Enfamil AA are two of them. All of these choices are likely costly and difficult to obtain, but may be simpler than making your own concoctions. It's up to you. Of the alternate "milk" products, you are right that Rice milk has little more than sugar, water, and the nutrients they add, which are good nutrients, but it's not complete nutrition. Soy milk offers the most nutrition of the options. While I'd have concern about a 100% soy diet for a newborn, there are many large studies on those who were raised on soy from birth, following up on them for several decades now, with only very tiny differences found in these children and those who received cow's milk formulas. Neither cow's nor goat's milk are complete nutrition either. The idea is that once a child is consuming other foods, they then get different nutrition from different foods. If it's not breastmilk or formula, their "white liquid" food should not be more than 12 ounces or so per day, to leave room for other nutrients.
So, for a short trial, you could go ahead and try rice milk, hemp milk, which has good fats, or another, just to see if your son improves. More nutritious options that you will want to use if he's clearly better without milk proteins, can be made in your kitchen. You can boil an organic chicken for most of the day and then dilute and use the nutritious broth, with a bit of apple juice in it. Baby food peas and some other baby foods can be added to rice milk, to increase its nutrition. There are rice protein products in the healthfood stores with a full spectrum of vitamins and minerals added, such as "All-One" Milk Free Rice Based protein powder with vitamins and minerals. (Watch out with that product though. They had another rice-based one that used calcium casienate -- milk protein. Hopefully they stopped doing that.) Spiru-teen is a good soy-based product, if you want to try soy. Get something that says milk-free. All of these suggestions, besides the formulas, are only for babies that are consuming other foods. You are right that some of these options are low in fat. It may be easier to attain fats in other parts of their diets, as sometimes oils added to a bottle only stick to the walls. Cod liver oil is one option.