you cannot take one home
A Bug Net does nothing here. Sea turtles fall in the same bracket as Dolphins, counted as too large to stuff into a container, so no cage item exists and there is no way to carry one back to base. The ocean ends up being the one biome you cannot decorate with its own wildlife, which is an odd gap given how much else in the game can be bottled and hung on a wall.
logging the entry
That leaves killing one as the only way to record it, and with the health it carries almost anything does that in a single hit. Do it on the same trip as the dolphin, since they share the same stretch of coast. Do not leave them sitting around either: hostile spawns damage them the same way you do, so something passing through can delete your target before you get to it.