Southern Rail still the best option?

Tower Bridge area on the south side

Very likely a Thameslink train to London Bridge station would be better. Tell us the name of the street and we can give you better directions.

With a group of 6 make sure you take advantage of the Group Save discount.

https://www.thameslinkrailway.com/tickets/discounts-and-railcards/groupsave

"Does Southern Rail run on Sundays? (In a post from a few years ago there seemed to be some kind of 'rail replacement service' since it was on a Sunday.)"

Track maintenance is usually done on specific parts of the national rail network at weekends, the post you saw was probably reference to work on that line at that particular time (it doesn't happen every weekend).

It is worth adding though, that if Southern Rail had a replacement bus service at that time, then it would almost certainly be the same for Gatwick Express as they use the same rail network.

A look at the Future engineering work page for National Rail enquiries suggests that there is work on the line, but everything should be operating as normal after 8:15 am on the 7th

For a group of six adults, why not price out the cost of a car service. It will take you right to your front door.

Thanks everyone! It looks like the group discount on ThamesLink will be the best in terms of time and cost. Thank you for the link to the group discount page and checking on upcoming maintenance.