Not all lactose free milk is equal. Look for ultra-filtered lactose free milk. The extra sweetness is caused by adding an enzyme that breaks down lactose into smaller, simpler sugars, e.g. glucose. Your body is better at tasting these sugars, and so you notice the sweetness more.
Ultra-filtered milk removes the brunt of the lactose before they add the latase enzyme, and thus there will be less simple sugar for your body to notice, resulting in a less-sweet milk.
Oat milk or water won’t add the body or depth you’re looking for. Depending on the particular beans, roast, and brewing method, milk might enhance the flavor profile. Similar to sugar and salt enhancing cacao. Yes, you lose nuanced notes, but no, that’s not a bad thing if the overall profile is improved.
You can also try blending different milks. Maybe you like 90-10 oat/whole milk or 70:30 ultra filtered/macadamia.
Perhaps go to a local shop during a slow period and ask if you can get a cup of coffee black and a flight of alternative milks. Whatever you do, have fun doing it!