This is a most interesting conundrum. I have discussed this with my husband and we have a some comments to contribute to your discussions.
Firstly, your instinct is 100% correct. Auctioning it could allow it to end up in the hands of a Neo-Nazi, who would love to get his hands on such an object. That would be a terrible thing to happen, so I don’t believe this would be the correct route to go.
It would be certainly be worth contacting the Imperial War Museum to ask if this item is of interest to them. It may be something they may wish to be exhibit, or to add to their archive for research use by historians.
If they were to offer to purchase it, it would be appropriate to donate the entire net proceeds to a suitable charity, like the National Holocaust Centre and Museum, which educates primary and secondary school students and university students about intolerance, bigotry, religious and race hatred, gender prejudice and bullying. Their whole being is to teach about the evils of such things, using the Holocaust as the most devastating example, to help the young people to stand up against such terrible crimes.
To profit from the sale of a knife, which belonged to a member if the Nazi regime who could easily have been a perpetrator of crimes against humanity and genocide would, in our view, be highly immoral.