Question first: Where is the owner getting the blood from? There are multiple sites, in dogs I believe the recommended area is inside the upper lip, but the outside of the lip would work, as would the paw pad, the calloused area on the leg, the chin area, a pinch of skin on the rump. Those may be a better choice for a stranger.
There's also the possibility of using an unused insulin syringe to draw up a little blood to use.
Secondly, if the worry of a bite is putting you off, is it possible to use a muzzle? Even if it's just a homemade one, like a piece of pantyhose or knee high wrapped around the muzzle. That would eliminate the worry, and it wouldn't be long term, so should be no trauma or stress for the pup.
Poor baby. It's a testament to your ability with dogs that your neighbor is willing to trust you with her. I hope you can work it out.