She definitely isn't exactly seeming stable. The blood on her face isn't contributing to it, but the stuff she's saying? Also not the greatest thing to hear in this situation, especially when Bigby is still so high strung, still determined to take care of any threat he can tackle here, since he apparently just can't do a thing about the Darkwalker as he is right now.

So the least he can do is make sure that Jessica doesn't end up hungry enough to just randomly attack someone in town.

"How much blood do you need to feel less hungry?"

He isn't crazy. He isn't going to get himself killed. But if a bit is enough, just enough to tide her over while he figures something out, maybe seeing whether or not she can't just drink animal blood-- he's got plenty of that, especially with how often he hunts.