I wouldn't worry. My son only took a bit to start with from the cup, now he happily drains it and would have more if I let him.
I'd persevere but I'd also include at least one main meal a day with milk as a sauce in the savoury bit, e.g. fish pie, cauliflower cheese, creamy pasta. That way you're getting even more milk in. If you then do a milky pudding (e.g. yoghurt, pannacotta, rice pudding) for tea and lunch, you're probably giving enough.
Have you tried carefully giving some directly from the cup without the spout? That sometimes worked for my son when he was first learning and you might find you can miss out the sippy stage.