I'm sure that you are correct in assuming that the word is related to a first-aid manoeuvre of some kind and I can't think of a perfect translation.  Interestingly however, during the Cold War, elements of the East German military used the expression when referring to the rapid evacuation of a military casualty.  The word used by the Brits (then and now) is casevac which is an abbrevition for casualty evacuation.  I know that casevac isn't the word you are looking for but it's the first time I have heard the expression Rettungsgriff since 1960 and I couldn't help sharing !