Now that I think on it, it's more likely as Alice says, that she has to be careful about endorsing her Muggleborn father by preferring his name to her given one.

As to the Order, well, I haven't had an owl from her in a while, actually, but I'm prepared to risk easing her in. Thus far she certainly hasn't balked at my contact or assistance, and everything she's done with and for Arthur suggests that she could be a valued member if she's given the chance.