They didn't make up the term "sea scorpion" for this find. It's the colloquial term for eurypterids, which are a well known class of extinct arthropods.
The stinger may be the most well-known feature of land scorpions to to lay people, but there are a number of body structures of sea scorpions that are similar to land scorpions. These features, and the general shape of the creature, led to the term sea scorpion.
The claw might be from a creature with outsized claws, but it's likely that it's from a creature with claws that are in the same proportion to its body as all the others they've found from the same species. This type of extrapolation is not unusual in paleontology, where they deal with incomplete fossils all the time.
And regarding the "arms race," note that body size itself is a weapon. That is, weapons don't have to be what we immediately recognize as weapons: claws, teeth, armor, stingers, etc. There were dinosaurs that developed what were obviously weapons to fight increasingly-large predators (triceratoops with the horns, ankylosaurus with the body armor and tail club, stegosaurus with the spiked tail) but the gigantic size of a brachiosaurus was a weapon as well. Similarly, a sea scorpion growing larger, but maintaining the same proportions, is also participating in an arms race.