Wow, tough one. Sorry to hear about your furbaby nearing the end. It's never an easy time.
I also think it was very kind of your friend to offer to go with you, but totally understand you wanting to go alone. If she's a really good friend she will understand. If you feel you can, do offer to do the same for her when the time comes.
With us, DH could not come with me to have Shadow pts. He said he would have a heart attack, he just couldn't do it. He stayed home with Halo and my daughter in law, while my son came with me. We both cried at the vets and then he drove me home. I think I could have driven myself home, but it was comforting to have him with me. This is a very personal thing.