I don't recommend anything i haven't actually done so my list is pretty limited. Beyond Budapest tour linked here was one that I did take and it was one of the most fortunate experiences of a dozen years of visiting Budapest. http://www.beyondbudapest.hu/english/tour_juice.html All of their tours sound extraordinary; especially the cemetery tour. This is a marvelous place to visit.
The tour at the Dohany utca Synagogue is also very, very good. Although I recently ran into the woman who had guided the tours for seemingly 100 years. She complained that she had been replaced by young non-believers. She was great, but she had some nationalistic views that might have caused her some problems. Budapest is an interesting mix of contradictions. Part of its charm. Besides just the synagogue tour you can book extended District Tours with them: http://www.greatsynagogue.hu/jewishquartertours.html#stand
Budapest is full of hidden or little known Jewish history and culture. This gentleman, while not Jewish, knows where most of the bones are hidden: http://www.guideinbudapest.com/ He is a private guide that I have worked with for 12 years; and hired myself on a number of occasions when I wanted to know more.
And I will send you a PM with one additional idea.