Vegetarian and low carb places to eat?

My partner and I are visiting from Austin this weekend, mainly going to the museums and parks. We have competing dietary needs, though – I am vegetarian, and he eats meat, but his diet has to be really low carb for health reasons (similar to keto, but I won't go into the whole thing here). We tend to gravitate towards Mexican and various kinds of Asian restaurants because that'll usually accommodate us both. Other cuisines are great too, though!

So ideally, are there any spots that check most or all of these boxes?

– Good vegetarian options

– Low carb options (or easily modified, ie a stir fry with extra veggies instead of rice)

– Outdoor seating and/or nice atmosphere

– Near museum district, the Heights, or Montrose (we will already be in these areas, but open to going somewhere else too)

Thanks!