Tested it out, there is actually a one turn cooldown, but the in game moves make no mention:
I tried the new move. It does have 1 cooldown during the battle, just is not shown in the description.
Sometimes, Ludia doesn’t consider errors in descriptions as bugs that need to be fixed. For example, in “Fearless Alert,” the description should be “Precise Attack” instead of “Attack.” I brought this up last time, but the rep simply responded that the move does indeed do precise attack instead of attack, so that’s that. The description still shows just “Attack” now.