I've thought some more about this and I'd say the knife might actually work.
We do know the tree didn't like it, at all. This will most likely also apply to all parts of the tree (i.e. the veins). The only thing I'm worried about is how much it might affect Yuyuko, given that she was a part of the tree until a few minutes ago. But still, I think it's our best shot. We did already cut her free from the tree and she appears to have survived it, so I guess getting the veins out should be a minor issue in comparison.
It of course might backfire (if she's dependant on a connection to the tree and getting the remaining veins out completely severs that connection, she might die; but if that's the case, I don't think there's anything else we could do, anyway), but... I think we should try it.