Well, the thing about the Foreign Legions and their offshoots IOTL is that they were almost universally designed to get around legal restrictions on the Army.
The French Foreign Legion, for example, skirted regulations against foreigners joining the Regular Army, allowed the monarchy to make its own independent army unit (a fear of some, may not have actually been causational) and alleviated some of the issues with using conscripted troops in postings like Algeria.
The Royal Dutch East Indies Army meanwhile, skirted some of the same issues regarding conscription for service in the colonies and made a framework for the use of native levies.
The problem is that, as of 1848, there aren't a lot of the same restrictions on the Prussian Army and there is as yet, no problem with colonial postings. I'm assuming 1848 was meant to tie in with the Liberal troubles across the continent, but OTL saw a rise in German provincialism almost as a counter-reaction to the liberal nationalism of '48. A multi-national/provincial unit would likely be seen as a fifth column, a breeding ground for radicals and unlike in France or the Netherlands, would serve no obvious purpose.
The most interesting candidate for a foreign legion analogue would be, in my opinion, a British one. But I can keep my thoughts for another thread.