First, I don’t know of any grass that is not safe for dogs.  It is typically the chemicals put on a lawn that may cause problems for dogs. Second, it is getting too late in the year for sowing grass seed as this is the time for the average first frost.  Grass that has just germinated will be more likely to be killed by a frost or will be stressed by the cooler temperatures.  However, as it seems like our summer is going to last and you have such a small area to seed, you could go ahead a try to plant a cool season grass such as fescue or Kentucky bluegrass.  Warm season grasses will go dormant after the first frost.  I’ve attached a link to a maintenance calendar for fescue and bluegrass, if you decide to try to plant now.

https://content.ces.ncsu.edu/tall-fescue-and-kentucky-bluegrass-home-lawn-calendar