How can I negotiate for significantly more than the listed salary range?

I hope I don't get shit on for asking this question because I don't know if what I'm trying to do is considered to be taboo or a bad idea.

I found a role that I'm super interested in at a company and industry I'd love to work in. The problem is, it's a pretty entry level role with an upper-level pay range that's about 60% of my target and 78% of where I am now. I reached out and it sounds like they want to talk to me (probably because it's so far below my experience level).

Now, I'm trying to figure out if it would be possible to negotiate for much more pay if conversations go well. It's a fast-growing startup with a lot of interest and investment, and thus probably some spare cash. I'd be open to (and probably prefer) an expanded role that is more mid-level than this initial posting, and I figure that's probably the best route if I want to get that pay up.

Is this a terrible idea? Is it possible? Any advice for making it happen? Do I mention this right off the bat, or wait until I've had a couple of interviews to say "actually this pay doesn't work for me"? I don't want to waste anyone's time, including my own. I also don't want them to look at me and say "you idiot, did you not read the job description?"

Thoughts?