Which Premier Inn in London to choose....

You might consider looking at the Premier Inn Waterloo, which is right up the street from County Hall. It's a little farther from Waterloo Station, but closer to Westminster Bridge, which is convenient for crossing over to Westminster and other sites. I stayed at County Hall but there are a lot of folks here on the form who have stayed at the Waterloo PI and liked it a lot.

"which most hotels don't accommodate unless we get two rooms"
I don't think this is true. Loads of hotels have triple or quad rooms. A triple might have three twins, but it might have a double and one twin. Likewise, a quad room might have four twins or a double and two twins. Sometimes "family room" is used. Here is an example from the Darlington Hyde Park, where I recently stayed:
https://www.darlingtonhotel.com/accommodation/

I know I recently used a site that allowed me to select how many beds, but I can't find it. Of the ones I tried, Expedia produced the best results when I put "three beds" into the "find your perfect stay" box. Of course, Expedia has loads of other search filters. I generally wouldn't book through Expedia, especially for an independent hotel, but I'm happy to use its information. Double check with the hotel that you're booking the right room.

Haven't stayed at any of those myself, just offering my 2 cents on the locations: The Kensington/Earl's Court location is walking distance to the V&A and the Natural History museums. The Southwark (Bankside) location is more scenic; you can walk to St Paul's via the Millennium Bridge (15-20 mins) or walk to the Tower of London via Tower Bridge (20 mins). And Borough Market is very close by. Both areas are safe. We walked all along the Queen's Walk/Bankside even in the evening without any issues.

Another approach to this challenge is to compile a list of Forum-recommended London hotels and check their websites to see which ones have a room to accommodate you on your travel dates.

Southwark Station will be a long way from anywhere of tourist interest. Not "bad", just a bit isolated.

I’ve stayed at both of those Southwark locations a few times for family trips and like them. There are plenty of restaurants and small supermarkets around. The locations are quiet. You’re close to the South Bank which is great for walking. The hotel near Southwark tube is very new and also a short walk from Waterloo for great transport connections.

We've stayed at both the County Hall and Waterloo locations. I'd say the two are almost equidistant to the bridge. I'd actually say CH was a quicker walk. We preferred County Hall, but would not rule out Waterloo.

I'll add there's a Gail's cafe in between both County Hall and Waterloo PIs...great for breakfast....