i dont think they care if they are legitimate killing is something that thrives in war and peace. The Brotherhood doesnt seem to be in a financial crisis and legitimacy will only hamper them with laws and restrictions.
Spoiler
[They're down to their last sanctuary in Tamriel at the start of their questline, and in an only slightly better situation at the end.They're down to a listener, a keeper, two assassins, and whoever they find to join them. They spent most of the money left in their coffers rebuilding the Dawnstar sanctuary, and will likely spend even more attempting to rebuild the order.
That is a pretty big crisis. If I (as Emperor) and find them, and offer them a deal, I can be fairly certain that it will at least be entertained, and almost certainly accepted. This deal would be legitimacy with limited restrictions (similar to the Morag Tong of Morrowind), monetary aid, and recruits, in exchange for training my Nighblades and swearing loyalty to the Empire.
Legitimacy would allow them to rebuild without fear of being exterminated if they are found out, and the money and recruits would help them rebuild.