Hello Eric,

These are difficult conditions for growing grass because it prefers full sun- 8 hours or more- and when stressed by too much shade it will always be thin.

Fescue type grasses are the most tolerant of shade. The type of grasses that tolerate heavy traffic- the ones used on sports fields- would be fescue , blue grass and ryegrass, as well. The types of grasses that handle traffic for sports fields are turf type tall fescue or a mix of Kentucky bluegrass and Perennial ryegrass. Tall fescue type grasses have the advantage of larger root mass which means better drought hardiness. For Tall Fescue (turf type) be sure to get a blend of several different tall fescue cultivars. Use of multiple cultivars can help prevent spread of disease and improves overall hardiness.

Some folks give up on the grass idea and put in alternative lawns mixed with perennial flower beds and mulched or paved pathways where the dogs regulary run.  Having a certified arborist come in ( see treesaregood link below) and assess the trees - giving you a plan on how to correctly thin them or raise the canopy- can put more sun on the grass, resulting in a healthier turf stand.

Finally, mowing, watering, and fertilizing correctly will result in the best grass that can grow under your sunlight conditions. Yearly over-seeding in spring can help maintain a thicker lawn in shady spots. 

The large dogs could be managed by sectioning off the yard ( perhaps with decorative fencing?) and alternating their access to each section, so as to distribute the wear on the grass. 

Here are links to best lawn practices and more to get you started-

https://www.canr.msu.edu/home_lawns/

https://plantscience.psu.edu/research/centers/turf/extension/factsheets/seed

https://aggie-horticulture.tamu.edu/PLANTanswers/turf/Dog_lawn_problems.html

www.treesaregood.org

A soil test to determine how much fertilizer is needed is always a good idea- 

https://www.canr.msu.edu/resources/dont_guess_soil_test_get_your_home_lawn_and_garden_soil_test_kit_today

If you have questions on any of this, please write me here and I will be glad to go into more detail.