V., my heart goes out to you and your granddaughter. In addition to what some of the other mamas have said, I don't think there would be any harm if you can show that your home is ready for a toddler. If she can have her own room or her own bed in a shared room, that's great. If you have pets, especially so called "bully breeds" like pit bulls and rotweilers, show that they are an unlikely threat, possibly b/c they are penned outside or crated. Do you have books, toys, etc. ready for her?
Happy pics of you and your granddaughter wouldn't hurt either, I think. It can help serve as physical proof that there is love and a bond between you. They will probably want to know about any other people that live in your house, so photos with them would hurt either, and be prepared to answer questions about them.
As Shelulu pointed out, second hand info about the boyfriend's sister probably won't hold a lot of water. If you know for sure of anything like a police record or that she's been evicted or had utilities cut off, bring it up. It shouldn't be that hard for the state to find out if it's true.
I have never been down this road but it seems to me like there is no harm in physically showing that you are prepared to provide and care for your granddaughter full time. This isn't about who has more money, although obviously being able to provide for her is important. To me, it's about showing that you are ready, willing and able to take on this very important responsibility. Sometimes a picture is worth a thousand words. Hugs and prayers to you!