I'm not sure if it's technically in Legaspi village, but the outdoor restaurants in Greenbelt 5 are typically occupied by dog owners. Mind you, these are typically smaller dogs, but they may be just what you're looking for if you want to being the whole family out for lunch or dinner (dog included)!
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Not exactly a restaurant but Legaspi market is open to pets! There are a lot of stalls with stuff for kids too (super cute plushie toys, tepees, cloth books, etc) so the whole family can have fun.