I'm glad you posted this. We know that Mycroft is genuinely "concerned" about his little brother but if Moriarty was being ironic in calling Mycroft the 'Ice Man' a good way to burn Mycroft would be to irretrievably tarnish the Holmes family name and in doing so possibly damage his official standing as a government consultant by making Sherlock appear to be a suicidal criminal  fake genius and sociopath.  There's another good reason or two that Moriarty would want to be rid of Sherlock anyway, for limiting of Moriarty's reach and doing the legwork that Mycroft is unwilling to do.

Hmm, how utterly ordinary.
The bad guy's enemy is the Government/establishment?
That's rather 'usual' really.
And quite boring; having Mycroft as an enemy. His hands are tied in certain areas, there are certain things that he would not do. Let's face it, can't see Mycroft standing in Buck Pal in just a sheet, can you?
But more sensibly, Moriarty sees everything as a game. All his crimes are just part of a big game, a game that he wants to win so that he can get the attention, the plaudits. A game that crowns him king of the world.
He doesn't care who the opponent is, whether it be little Carl who laughed at him, or the Establishment who can't keep security in prisons under control. They are just all games.
And he knows the only person, the ONLY one who would be willing to rival him is Sherlock. Sherlock is the only one who would forsake legalities or morals to battle wits with him.
As he says himself, he may be on the side of the angels ... etc.

And that is the only worthwhile rival for Moriarty; anyone can rival the likes of Mycroft.

Yes, undoubtedly the Holmes Brothers want the same ends.
Let's look at the first main confrontation with Sherlock at the pool. Now had that been Mycroft rather than Sherlock there wuould have been no witty repartee, no challenges being thrown down etc. Mycroft would hav just arrested the dude! Game over.

The British Government doesn't enter into power plays with criminals, they make the rules, they change them when needed. Either way, it's Game Over, ' we won't play ball with you'.

And that, to Moriarty is a dull and boring contest.

if Moriarty's rivleal was  mycroft why would he want all the infomation about sherlock unless it was to kill sherlock then get to mycroft