Yep. It happened to us with Malfi.
I just got a cottonball and alcohol and wiped the area.
It didn't really bleed much.

You are doing a great job, don't worry so much about the pain. I think it only stings for a second, like when a human gives blood.

Sometimes I have gotten so frustrated I just want to 'pay the vet' and not worry about it. I figure, just as long as he gets what he needs, no matter how it gets done.

I feel for you, we are lucky, we are not doing fluids now, but I know they are in my future.