OP - I've recently posted a thread about my beloved dobermann that I had to have put down due to her going for my son.
BUT she was almost 8 and we'd worked with her for years, so I hope that you will try with Jock before giving in.
Firstly as others have suggested a behaviourist may be able to help diagnose why he is growling, however as he's had 3 homes it sounds like you will never know the truth of his past.
Secondly - get him muzzled now, whenever your daughter is in the house, until you sort this it is not worth the risk. Despite him being small he could scar her for life (or worse) in seconds. My Dobe being muzzled is the only thing that saved my son from serious injury (as it was he ended up with a black eye, a lump on his head and a bloody nose). Soft muzzles are not going to cause him much problem although he won't like it, and you can take it off when she is at school / in bed etc.
Jackies are known for being snappy, and their size doesn't help - they tend to think they are big dogs.
i hope you work it out. But as I've discovered its best not to take chances, you need to make sure your daughter is safe first off.