So I ended up doing the battery replacement myself. It was not easy at first, but once one understands how things operate, then it is OK.
Unscrew the case back, manage to remove the used battery (the hardest part) somehow without destroying the whole thing, slip in the new one, grease the seal with silicon oil for waterproofing on an earbud, reset the watch, screw the case back, check the ringtone and alarm, and voilà .
Total cost £4.5 for the batteries (1 spare left), borrow precision screw drivers from the first cousin, use the silicon oil from the speedcubing days, and the watch is as good as new.