If he only had, all the subsequent events would have been avoided as Dumbledore would probably have solved the whole case earlier... We know Harry writes to Sirius right away to tell him about Crouch sneaking into Snape's office, we know they discuss it at length in the mountainside cave and we know that Sirius is in close correspondence with Dumbledore too. But we know for sure that Sirius did not warn Dumbledore: When Dumbledore interrogates Crouch with Veritaserum, he turned up to be completely unaware of the existence of the Marauder's Map (Crouch mentions the map of Hogwarts he took from Harry Potter, and Dumbledore answers "The map? What map?").
So how come Sirius did not write to Dumbledore to let him know of that highly bizarre fact?
It seems all the more anomalous that Sirius gets back to Hogsmeade precisely upon learning this very fact from Harry's letter. Not to mention the fact that it is Dumbledore himself who designates the mountainside cave as the safest place for him to hide! And finally, the way Sirius discusses the matter with Harry, Ron and Hermione confirms that he finds the information highly suspicious (Barty Crouch is supposed to be badly ill and, as Sirius points out, staying at Hogwarts to judge the Tournament would be the perfect excuse to keep an eye on Snape or anyone else)
The only 2 hypothesis I can think of are not convincing at all:
Sirius, Harry, Ron and Hermione are all preoccupied by the fact that Dumbledore seems to trust Snape, and Sirius doesn't want to confront Dumbledore about this; but he could nevertheless have let him know the simple fact that Barty Crouch was seen at Hogwarts!
Sirius doesn't want Dumbledore to know the existence of the Marauder's Map; but why would that be, since Dumbledore already knows by then that Sirius became an Animagus while he was at school? Moreover, the map would be very useful to Dumbledore under such circumstances as those in the Goblet of Fire...
EDIT: I just thought of another hypothesis and I reckon it's the right one: because Sirius, trusting Moody as everyone else, and knowing that Harry had told Moody about seeing Crouch on the map, would naturally and obviously trust Moody to tell Dumbledore about the fact! Besides, is Sirius not convinced that Dumbledore hired Moody precisely to have an Auror in the castle keeping an eye on dark wizards like Karkaroff? So, of course, Sirius would expect Moody to tell Dumbledore, not knowing Moody is an imposter. However, it remains odd that, in his further correspondence with Dumbledore, none of them ever mentioned this Crouch business.. Which would have led to Dumbledore realising that Moody was holding some information back! And starting to suspect him...
What do you think?