Dogs and fescue are a problem. Fescue is a bunch type grass that must be reseeded every year (in the fall) until you have it dense enough to keep weeds out. Then you have to evaluate it every summer to see whether there are thin spots that need more seed in the fall. Kentucky bluegrass was the spreading type grass. It must have full sun to thrive. You might get some to grow but with dogs and in the shade it will not perform like it would under better circumstances.

Dogs and shade are a killer combo. There is no combination of grasses that will do what you would like to do.

You are doing the right thing by wanting to start early, but spring is actually late. Fall is early. Spring seeded grasses tend to weaken and die in the summer heat leaving your lawn open to crabgrass. With your shade any crabgrass that gets started will not be very strong. You might even have too much shade for crabgrass. That would be refreshing.

I would suggest you make do for this season and put most of your budget into improving the yard in the fall. Practice good watering (deep and infrequent), good mowing (highest setting for fescue), and good fertilizing. If you want to use chemical ferts, apply later but no later than Memorial Day. If you want to use organics, you can start 3 weeks prior to your last frost date and use as much as you want. More is better. If you want to seed now anyway, keep in mind you will be seeding again in the fall to help 'densify' the turf for next summer.