I know it's not the same as a commercial airliner, but I used to teach hang gliding on the beach and we'd sometimes get a little packed sand in the back of the keel (which is the name for the central bar that the wings were mirrored across). The total weight of the student plus the glider could get up to ~280 lb, and the amount of packed sand would be less than a pound or two, but the flight characteristics of the gliders would change drastically, to the point where we could barely get them in the air.
As another commenter notes below, the few people moving around the cabin is less about control ability and safety than it is about fuel efficiency.
It is not about balance.
Moving a few people probably doesn't cause an issue if the rest of the plane is completely empty.
If you move 20 people from the right side to the left side of the plane this will probably have some kind of effect.
On smaller regional aircraft they sometimes ask people to move seats to the other side or back to front.
They rarely upgrade to international first/business because the meals and other perks cost real money. And it'll piss off the people who payed over a thousand dollars for those seats.
That's probably a mix of people with critical jobs needing to travel, people having to move for a variety of reasons, and people listening to the ignorant and ignoring the advice to social distance.
It's better for them to run highly unprofitable flights than miss their contract obligations with big corporations that want a plane available to move people transatlantic on a day's notice.
Had an empty flight with Singapore Airlines right after Christmas and they didn't.
This is British Airways so we'll see.
2. Does it cost them significantly more to seat you in a different location? I'm asking, I genuinely don't know.
Right now I'm in premium economy as for some reason the ticket was cheaper than economy.
Realstically speaking there is virtually no difference in service for them.
Assuming that I get the usual economy food and I'm really the only person on the flight.
Now, if you fly regularly and have some miles status with that airline you can request all sorts of things. The gate agent has a lot of leeway in that regard.