Going to London and Paris back to back - trying to add Bruges - Any suggestions?

If you've got the time, I'd definitely add Bruges afterward. It is a pretty easy train ride from Paris. You have to change trains in Brussels but there are usually hourly trains to Bruges. You could plan to fly home from Brussels OR hop up to Amsterdam and fly home from there, depending on where you live and what flights work for you. I'd want to spend 3 nights in Bruges if you can - that gives you 2 full days. I could easily spend 2 full days there or you could use one of the days to visit Ghent. I prefer Bruges over Ghent although there are many of the forum who prefer the other way.

Which tours are you doing? Whatever tour you are doing from Paris, add days in Paris afterward. Giverny is an easy day trip even for someone like me who does not speak much French. You take the train from Gare du Nord to Vernon, then when you leave the station there are buses that take you to Giverny. Last time I went it was about 8E for a round trip on the bus, pay the driver. The buses are painted with Monet's water lilies so it's pretty easy to figure out, lol.

I would also suggest you arrive in London a few days ahead of your RS tour. Give yourselves time to get over jet lag and to allow some flex in case of transportation hiccups.

Pam is always right, except this time she missed the station where you get the train for Vernon. It's Gare St-Lazare, not Gare du Nord. Gare du Nord is where you'd get the train for Belgium.

It is indeed easy to get between Bruges and Brussels, but you have to plan in advance for the lowest cost trips between Paris and Brussels. In most cases, a ticket that says it is good to Bruges would be valid on any of the local trains from Brussels that day. There is a ticket machine outside the Bruges train station for the busses to downtown. This is very much a spur journey, so you should not assume it's as efficient as a single-seat ride somewhere. You can't go to Amsterdam from Bruges without returning to Brussels on the train. It would be tedious to return to Paris just to take a flight home.

These unreserved-seat local trains are fine, but they may not have long distance sized luggage racks, and may have double-decker commuter cars that are brutal with huge suitcases. Pre-pandemic, they were very crowded at rush hours and school hours. BRU airport is a separate train from Brussels, although I think there may be a tiny number of direct trains for Bruges (only?) from the airport, due to demand. American airlines tend to have remote, bus gates at BRU, with a very long walk inside the terminal, and a long line for non EU passports. (Pre pandemic comment.) Airport taxis are very expensive in Belgium. The NMBS rail site is very good, as I remember.

Depending on how long you want to be away from home, I'd consider a future trip instead, to combine this part of Belgium (say, Antwerp, Gent, Brussels, Bruges, Leuven, all from a base in one of those, my choice Antwerp) with something else like the Netherlands or western Germany. For example, Cologne and Dusseldorf are an easy train ride from Brussels.

I can't imagine spending the night in Giverny. It's such an easy day out. Advance attraction ticketing may be essential by now.

Dick!!! Thank you for the correction! Yes, of course it's not Gare du Nord for Vernon!