Staying in London - Day Trip to Both Canterbury and Dover?

I think if you did a circle trip: London to Canterbury then to Dover, then back to London you could make the logistics work. However, I think only one day trip per day would be better. What do you want to see in each town?

"Two different directions" is an overstatement. There is a train route to Dover from London that passes through Canterbury. Dover and Canterbury are connected by a 30-minute train ride. I would question whether Dover is worth visiting at all, but that's just me. You can certainly combine these in one day trip, assuming you don't have a full day's worth of activities in each.

A lot depends on what you are wanting to do/see in both places.

If you are wanting to visit Dover Castle, it is recommended to allow 3-4 hours minumum for the visit as there is so much to see there. In fact it could easily fill up the full day.

Ditto Canterbury. If all you want to see is teh cathedral then a visit will take about 2 hours (add on extra time if you want to include the guided tour) If you have plans to do anything else in Canterbur, then again you could easily fill a day there.

I'd want more time than that at both Canterbury Cathedral and Dover Castle.

Ideally, I'd like to leave as early as possible, visit the Cathedral, have some time for lunch and exploring the historic center of Canterbury, then head down to Dover to visit the castle and stroll on the cliffs at twilight. But I am increasingly thinking it would be too much for one day and I should simply focus on Canterbury.

It really depends on your interests. It would be easy to spend a full day at either of them.

Interested in Dover Castle and the war tunnels? Then Dover. Interested in the Cathedral, walking through a busy shopping area, or punting on the river? Then Canterbury.

am increasingly thinking it would be too much for one day

You have summed it up accurately here - your plans will be short changing both.