A 77 Sportster sounds like a winner to me. You could file for a lost title and go from there.
I have a 1975 GL1000 Goldwing in my garage that I was going to turn into a rat bike. I disassembled it and left the frame outside so it could rust. My plan was successful, but I dunno if I will ever get time to finish it. I was going to make it look something like this: http://4.bp.blogspot.com/-4qnrMzfggCc/TtaZybHmziI/AAAAAAAAGHs/1sFtm9ekkAE/s1600/goldwing-rat-bobber_mondo-lulu-photo.jpg
If that interests you, then I could work something out on it as well. Not only that, but a price budget of $5-7k is REALLY high for a rat bike, Sportster or not. If you have that budget, then you could build something NICE, not rat. If a Harley is what you want, then stay persistent with it. If you decide that a non-Harley would be fine, then let me know and I can give you all kinds of ideas with that budget.