If it helps you, I'm your same height, about your same weight, my lifting numbers are about the same as yours, and am going through the same ordeal trying to gain. I started at 145 and am now 176. Did gain some fat like you did - not concerned about that at all. When I started i couldn't eat more than 2100cals comfortably. I decided to stretch my stomach out little by little. I chose one meal to make bigger each day - the first meal of the day. The other meals could be smaller. The one bigger meal scheme worked tremendously better than spreading the calories out evenly throughout the day. Stretching the stomach is key. I'm up to 3000cals per day pretty comfortably.
I also have dairy issues like you, plus grain issues which makes it a little harder. For dairy sometimes I substitute things like almond butter: low-volume, high-calorie. Sometimes I take a lactaid and eat cottage cheese anyway (which has low lactose to begin with). Goat's milk is also better in terms of bloatness and lactose intolerance. If I'm having more than 8oz, I have it with lactaid anyway.
Hearing everyone's advice has been very helpful for me as well.