As struth says, Chester is a wonderful place to visit – the pedestrianised town centre has Medieval galleries, there are interesting clocks on the arches at the original city gates, and it’s possible to walk along the top of the city walls though I have no idea what the wheelchair access for that is, or where. The are the remains of a Roman amphitheatre on the southern edge of the city (also ‘sort of’ visible from the car).
An hours drive south of Wrexham there are the remains of Wroxeter Roman city – mostly a big field that the dog will like, but the exposed foundations are surrounded by flat concrete suitable for wheelchairs. It’s not a huge place and you’d probably have to be a bit of a Roman enthusiast, but it would be a nice drive in good weather. Best take a look at the site on Google Maps to see if there’s enough there to be worth the time.