Note that the existing batteries are spot welded to the nickel plate joining the batteries to the PCB. I cut and pulled apart the nickel plate, it was sort of mangled, from the battery rather than desolder where the plate joins the PCB.

It appears you have or purchased the replacement AAA batteries with the nickel plates included.

Mine didn't have the nickel plates that join the batteries to the PCB.

I added some solder to both sides of the batteries and then each nickel plate, for each battery, pressed the batteries in place so the cold solder touched and then reheated it so the solder on the batter and plate joined.