For a bit of a bargain, Keswick Youth Hostel do breakfasts for non residents, next to the river opposite fitz park. I think the George does breakfasts for non-resi’s too, and thir grub is usually good. Not tried it, but another convenient one would be the new spot opposite WH Smiths, used to be the Rembrandt (Lakes Bistro?).
Left field suggestion, Nichol End Marina in portinscale, Random service from Roger but a fine place to be, especially on a sunny morning.
Living here and having 2 young kids, don’t get out for breakfast very often. Also, if you find yourself in Cockermouth, I very much recommend the Coffee Kitchen, and you can bring your bike in and leave it downstairs while supping/munching upstairs. Top spot.
Edit- another thought, Cranstons at Penrith, nice cafe upstairs.