Ah, S.. My heart is with you! I read your previous post and it hit a chord with me, as I too have been going through major best-friend (or shall I say ex-best friend) drama. You got such good feedback before that I ended up not posting a response.
I think, if you can go to a CASUAL lunch, just for the sake of supporting her in her difficult time, then go ahead and go. But I would keep it short. Lunch, and that is it. Express sympathy for her father's passing, ask her if there is anything you can do to help out, but LET GO of any hopes for a deep meaningful friendship. Cuz if you don't, you're just going to be back in this same boat in a few weeks or months: feeling let-down over this one-sided friendship.
It is so frustrating that the minute you are ready to let go, your friend pops in! That is what has happened to me over the last 8 months: reaching out, being rebuffed or ignored, and then the minute I finally am like "AHA! I get it! I am done and ready to move on!", that is when my friend has suddenly texted me or emailed me. Unfortunately, what happens is, I get all excited, open up my heart again, only to get it broken. Over and over. I know, sounds dramatic, but my friend really was like a sister to me and it has been devastating to let the friendship go. I am still working through it- but it does help to fill your life with new things and people. Good luck! I wish you the best.