There's your bait and switch, your cash and dash, your physical threats from her or an accomplice, your big dramatic scene, your blackmailing once she has your info... the list goes on. Arranging to meet in public before retiring to a more private location does nothing to reduce your risk of running into any of the above -- or LE for that matter.

What will? You already know the answer to this: meeting with someone who is well-reviewed or is otherwise highly recommended by someone you trust.

That said, many ladies love lunch/dinner dates, and may even offer a particular rate for it. Others aren't into it, and will balk at the idea, for their own reasons (privacy concerns, don't enjoy talking that much, are breathetarians, etc.). If it's not clearly stated either way, it definitely is okay to ask! Be sure to clarify, as you have here, that you do intend to compensate her for the public time as well.

I don't think that a lunch date is a good idea if you are just trying to eliminate the fear of LE...it is effective, I can guarantee that no LE is going to sit at a lunch for an hour, and then migrate to "play" time to catch a guy soliciting.  It brings them into uncontrolled environments that no LE would ever go into not to mention the waste of time.  There are easier, less expensive ways, to substantially eliminate LE concerns.

With that said, meeting someone for lunch prior to a P4P engagement is a great way to establish a connection and, for me, has always led to better play time.  So, if you are going to do it, do it to enhance your experience not to screen LE.  As it has been said elsewhere, most providers I've met love that idea and do offer decent pricing for a lunch/dinner date.