As I started reading your post, I though The Mayflower would be a good recommendation for you. And then I read on and see you picked it yourself!
I went there a couple of times with a family group last May and think it's a very good choice for you. It's rich in history, family friendly with welcoming staff catering to tourists without being a tourist trap. and locals go there too; you may get chatting and make some friends.
It's cozy inside and the outside deck is over the river at high tide (although it may be a bit chilly in November) and you'll get some nice views while enjoying really good food and beverages.
And if you want to combine it with a bus ride that passes some of the sights, take the Number 15 -- ideally one of the traditional Routemasters for an added touch of flavo(u)r -- from Trafalgar Square, via St Paul's Cathedral, to the Tower of London.
This link gives you more information. The buses run in both directions.
https://www.insightguides.com/destinations/europe/england/london/cultural-features/old-london-bus-tour-routemaster-bus-no15
You'll know what you can manage on your first day, but I think you have a fine plan and you'll certainly want to eat. If it doesn't work out, you can always go there another day.