If you really insensitive to lactose then yeah its going to be very painful, milk is in just about everything baked or with most sauces that isn't stamped vegan. At least most reputable places will take it seriously and have a proper allergen book.

I am Coeliac, and its like me going to Japan, just about everything has wheat added to it. Soy sauce? Gluten. Miso? Gluten. Whats annoying is that traditional Japanese recipes for Miso and Soy do not use wheat, it was added later after the American occupation. You can buy both soy and miso gluten free outside of Japan very easily, but in Japan, even though they made by Japanese companies? Ha good luck.

The worst part is that nobody in Japan takes it seriously as there been like two people in the last five years who were diagnosed with a gluten intolerance let alone Coeliac, so even if you take a Japanese speaker along and they explain it politely to the chef, you still get gluten.