It can be difficult to get seed to establish in existing lawn, and while it is doable, it can be tricky. Foot traffic from dogs can compound that even more. One option is to plant landscape plugs into your lawn. Plugs usually come in trays of 38-50 plants and essentially are baby perennials. If you can't find them locally, a number of nurseries you can find
online will mail them to you.
The advantage of using plugs is that they can potentially compete against the existing grass and withstand an occasional paw. You can plant them right into the grass but the less competition the better so you might want to consider eliminating the grass beforehand. Either way, loosen the soil some around each plug when you plant them so their
roots can start growing more readily and then don't let them dry out until they're established. Also, if you can keep them from getting shaded out by any lawn grass as it grows, that helps enormously.
If you were to take these steps, and the species you choose match the conditions of your site, the results using landscape plugs can be relatively fast; you can even have meadow plants flowering in your first season. Once the meadow plants establish you can then let the grass grow up because the species you planted from plugs will be able to hold their own.