First of all - make sure you press a pillow against your stomach when you sneeze or cough. It really makes a big difference. In terms of the milk, some people develop an intolerance to lactose after surgery. I think it's more common for bypassers, but I have heard of it happening to sleevers as well. I don't think I've ever heard of it manifesting itself as a cough though, but if that only happens to you when you have something with milk protein in it, then clearly there is a connection.
Perhaps try making your own shakes with soy protein powder in it. I make a home made shake (using a hand-blender) based on bananas, blueberries, almond milk and protein powder - but play around with whatever flavours you like. Berries are usually your best choice though since they are lower in sugar than fruit, and also have a lot of antioxidants. I usually freeze the bananas and blueberries beforehand and only use a little almond milk so it ends up like a thick milkshake, but since you are newly post-op I'd keep it as runny as possible. Perhaps substitute the almond milk for soy milk as well to maximize the protein content.
If you're not keen on soy, goat's milk also tends to be easier to digest for people who are sensitive to lactose. There are also other non-lactose protein powders like pea protein you can use instead, but these are incomplete proteins and lack certain amino acids.
Also, things don't have to be "milk" based - clear soups, broths etc are also usually fine at the liquid stage for most providers.
It's very common to feel weak this soon after surgery. I could barely get anything at all down for the first couple of weeks.
