I spent 3 nights in York and 4 in Canterbury in late May. I also spent 3 nights in Hastings. That gave me 2 full days in York, 3 in Canterbury and 2 in Hastings. I used trains, buses, taxis and feet to get around.
The weather in both areas was rainy, windy and cold most of the time. I would expect it to be the same when you are there.
I attended evensong services at about 17:30 in both cathedrals. I didn't go to Dover, largely due to the weather. I did go to the Hastings Battlefield, did the short walk around and just made it back to the café in time to order and get seated when a deluge hit and drove lots of people inside for shelter. You can also do a long walk which puts you right on the battlefield instead of just looking over it from the ruins of Battle Abbey.
Here's what I think.
You cannot really see the area around Canterbury, including Dover and the 1066 battlefield on a day trip from London. With a car on your own and with the short distances, you might be able to see more, but parking will be an issue. Also, the places you'll want to see do keep regular hours, usually closing about 17:00, and the days will be getting shorter when you are there. With the right train or bus scheduling, you might be able to visit Canterbury Cathedral and Dover OR the Hastings Battlefield from London, but not all 3 in one day.
Although some of my British ancestors came from Kent, and no doubt were in Canterbury Cathedral centuries ago, I preferred York. Besides the evensong service, I attended the York Minster Mystery Play one evening. I thoroughly enjoyed exploring the Shambles and the Yorkshire Museum. And I definitely ate at Betty's. I wish I would've had an extra day in York in spite of the weather.
So, to answer your question, yes, I would trade Canterbury/Dover for York. And to answer a question you didn't ask, I'd skip Canterbury/Dover and keep that day for London -- especially if I hadn't been there before.