From what you're describing, it sounds like a purposeful bite - the dog wanted to keep the napkin and used the amount of force necessary to make your daughter back off.
Kids (under at least 10), IMO, should not be unsupervised around dogs. It's too easy for something like this to happen. It's also worrisome that you say your daughter plays rough with her. If she's doing things the dog doesn't like - even if they aren't abusive - it's potentially damaging the relationship between daughter/dog and teaching the dog some bad habits. Both kids and dogs need to be taught how to coexist and treat each other with respect. Playing "rough" is not that respectful, IMO.
Many dogs have issue with stolen objects being taken from them. (Do a search...) I'd suggest a new house rule -- if puppy gets something she's not supposed to have, the kids must COME GET YOU and you use it as a training exercise to practice happily relinquishing prized objects on request. Better yet - she won't have to come get you b/c there would be little to no unsupervised time between them. By keeping them supervised, you can not only prevent unwanted behavior on both their parts, but can also reward the dog for being tolerant of low-level kid-induced annoyances and/or making the right decision to get up and leave when feeling uncomfortable by a child's handling, etc. BTW - "rushing" a dog is often seen as a sign of conflict and can put many dogs on the defensive.
Your situation reminds me of cases where ... after a bite ... it comes out that the kid in question was doing stuff to totally annoy the dog while unsupervised. Not abusive in the least, but if the dog doesn't like it, he doesn't like it... and will only put up with so much before using his resources (growling ... hard stare ... lip curl, etc. ... then the bite) Kids are going to totally miss the warning signs, so you end up with a bite.
(Golden Retriever) Born 5/12/12, Adopted April 2014.
Wordsmith's Winsome Warrior
No titles (I'm too lazy now!), but an awesome hiking partner! "Quiz"
(Golden Retriever) 12/31/03 - 3/20/13
U-CD Tanbark's Number Two Pencil, JH, CDX, Can-CD, AX, AXJ, OF, RS-E, JS-E, GS-E, WCDex. I loved being this dog's partner! Watch him shine! "Zoie"
(Whippet) 9/8/00 - 3/28/14
Donmar-Valair Second Wind, CD, RE, MC, ASCA-CD, Th.D.
#1 AKC Obedience Whippet for 2004 / AKC and ASFA FCH Pointed