I live in Richmond, although have never been out for breakfast or brunch.
My local knowledge would suggest the following places -
Morro Lounge (definitely dog friendly) and located right on the Market Square.
Holly Hill Inn. Also dog friendly, about a mile out of town. Pub type place, beautiful views from the garden. Very much a 'locals' type pub, but the breakfast gets good reviews.
Granny's Kitchen - very central, excellent local reputation, but dogs only allowed in the (sheltered) courtyard.
Middleton Lodge - very 'high brow' type place, dogs allowed only in the snug. Roughly a 20 minute drive from Richmond. Has an excellent reputation (possibly a Michelin star?)
Little Drummer Boy Tearoom - another lovely local place in the centre of town.
Ellerton Lakeside Cafe - 15 minute drive from town (but heading away from the A1M direction). Dog friendly, out in the countryside.
Mainsgill Farm Shop café - right next to the A1M, so possibly the most convenient for you? Lovely local food, and a large hall of shopping.
I'm not sure of booking arrangements. Possibly you'd need to book Middleton Lodge, but the others are likely unnecessary.
Enjoy North Yorkshire!