I agree, I would not let BBF use the sparkly backpack. At all. Give it back to MIL with a blunt 'you know my parent's are giving her one, don't ruin their gift.' Otherwise you will just teach MIL that she can go around you directly to BBF and that you won't stand your ground because you (rightfully!) don't want to upset your daughter, who has nothing to do with any of this. She'll just keep giving the inappropriate treats/presents/privileges directly to BFF to circumvent you, knowing you will let her keep them now that she's seen them.
Give the backpack back and every time she tries to go around you, stand your ground. Grandma says you can miss the nap? Too bad Mommy outranks her. Grandma gave you ice cream? No sugar in the afternoon (generic example here) and into the freeze for dessert it goes. BBF will learn pretty quick who's rules take precedence.
Of course, this only works if your DH is also onboard and backs you up.
It only works if she does it in front of me. When DH or I are there, she's very good about following our rules (or asking us first, if it's something she's not sure we'd say yes to). So she'd never suggest missing a nap or getting surprise ice cream without checking with us discreetly first. It's just when she's the one watching Babybartfast and she's using her own judgement that she tends to stretch the limits
If we said "Babybartfast must have a 60-minute nap, staying on the bed, with no toys, alone in the room with the lights off," she'd have no problem doing it. Same with "no sugar ever." But those are two rules we're context-sensitive about - Babybartfast needs a nap for 1-2 hours, except it doesn't always have to be at exactly the same time and she doesn't *have* to sleep and she's allowed some types of toys (books, mostly) but not others (iPad) and if we have something else scheduled she misses a nap occasionally and if she's in the right sort of mood, it's fine. I just can't trust MIL to come to the same conclusions about where on the scale things fall as I do.
I will say, at the restaurant the other night, Babybartfast opened each present, then enthusiastically ran to the person who gave it to her and gave them a big hug and told them thank you. That was all MIL's influence - I certainly would have encouraged the same behavior, of course, but apparently MIL had given her a little talk earlier in the day about appropriate birthday behavior and it stuck. So she's not all bad, just manipulative and lacking in common sense