"The Grin" does not refer to anything in Harry Potter. It is not mentioned in any of the books. Here is the scene from Prisoner of Azkaban in which we are introduced to The Grim:
"My dear," Professor Trelawney's huge eyes opened dramatically, "you have the Grim."
"The what?" said Harry.
He could tell that he wasn't the only one who didn't understand; Dean Thomas shrugged at him and Lavender Brown looked puzzled, but nearly everybody else clapped their hands to their mouths in horror.
"The Grim, my dear, the Grim!" cried Professor Trelawney, who looked shocked that Harry hadn't understood. "The giant, spectral dog that haunts churchyards! My dear boy, it is an omen — the worst omen — of death!"
The reference in your question is not to the Harry Potter books at all. Rather, it is the parallel scene in the Prisoner of Azkaban movie. As is common in the movies, a bit of license was taken to add/remove/change some things from the books. In this particular case, someone apparently thought it would be fun to throw in a joke here. To that end they had one of the characters mishear "the Grim" as "the Grin", and another character call the first one an idiot. Note that the second character does not even exist at all in the books; he is entirely an invention of the movies.
As your question is premised on a nonexistent character saying something in a nonexistent scene, it is clear that it has no relevance to anything in the Harry Potter storyline. It is theoretically possible that the movie-writers could tie the joke into something else, but there is not the slightest indication that they ever did.
Thus, to answer the question of whether "Grin" could be short for "Grindelwald", it would not be a strict logical impossibility, but as "the Grin" is clearly a joke, and has nothing to do with anything in the books, there is no reason why it would be short for Grindelwald.
Similarly, there is no reason why "the Grin" would be in any way related to Ignotus Peverell or the Invisibility Cloak (and unlike "Grindelwald", these two don't even share the letters g-r-i-n). In fact, Ignotus Peverell was not even introduced until the seventh book which was published three years after the Prisoner of Azkaban movie was released, so the writers wouldn't have even known about Ignotus Peverell to reference it.
There might be a slight commonality between The Grim and Ignotus Peverell in that they both loosely relate to death (the former being an omen of death and the latter being one of the three brothers who defied Death), but there is never any connection drawn between the two in the books. Indeed, after Prisoner of Azkaban the Grim is a non-entity, mentioned only once in passing in Deathly Hallows.