You are required to isolate with covid at home for 10 days and it is illegal to go out. At all. Unless fleeing DV (or presumably house burning down) or for urgent medical attention. "we want to walk our dog" really isn't a good enough reason.
So whatever challenge it takes, you really ought arrange dog walkers or ask on covid line - if councils are still doing them - for help to walk a dog. You need experienced dog walker as DDog is in season
Don't know why you're not asking friends and family here OP? - even one hour long walk a day would be something.
I don't understand why you would chose to breach Public Heath legislation, risk a fine and also risk others health when you know you are infected with covid19- an airborne contagious virus.
Whilst you may not meet anyone whilst out, if you had an accident whilst out - how easy it is to slip, a car driving too fast near you or others children barrel round the corner on scooters - it's still fire service and paramedics that have to help you. Still A&E you'd be going to. And you have covid. You never think it will happen to you but then it does.
Early morning and later evening is often when CEV people avoiding others may be going out to reduce their risk of infection.
Stay home when you have an infectious disease. No excuses.
I know it's not you saying you'll go out OP and you initially asked for advice on what to do , it's terrible some MNers on here are encouraging you to do something illegal and dangerous.