Hi OP, I live in Cumbria :)
The whole of the Lakes (and Cumbria) is really dog friendly and your dog will pretty much be allowed everywhere except food shops/supermarkets.
There's loads of great walks around Derwentwater and dogs are allowed on the boats too. I would recommend afternoon tea at Lodore Falls Hotel and Spa - it's dog friendly and the food is incredible - on a nice day you can sit outside too. You can get the boat from Keswick, explore the waterfalls, have afternoon tea then catch the boat back again :)
Be really careful with allowing your dog in Derwentwater though as there's often blue-green algae which can be fatal.
I'd take a drive to Ambleside and Grasmere too - the gingerbread shop in Grasmere is incredible and well worth the queues. In Ambleside, go get a picnic from Rattle Ghyll Deli (the food is vegetarian but SO good) and take it to the park, or get fish and chips from the Walnut Fish Bar.
If you want to go to the coast, drive down to Silecroft - miles and miles of deserted beach. You can park at Silecroft, walk along the beach to Haverigg (at low tide) and catch the train back again, or walk along the new England Coastal Path which has just been completed.
Silecroft beach is 100% dog friendly and you can let your dog run with no worries about sheep or cattle, though there will be horses on there at times!
Have an amazing time! :)