Let’s cut to the chase: generic
Chances are, your client is too busy for catching up. And they don't need you to buy them lunch.
Instead, place yourself in their shoes. Imagine that you're a "buyer" in the marketplace. What are your biggest challenges? What kind of information would help you to be more successful? And what solutions would you be eager to discuss with a trusted advisor over lunch?
This exploration will give you insight into an effective email to your client. "I've been giving some thought to X and how it might impact you in the coming months. I have some suggestions that I think you will find helpful. Let's discuss over lunch. How does the week of the 9th look?"
Now your email is a value proposition with a clear call to action. And you've provided a specific timeframe for their calendar. You will find that they are much more likely to respond to your email. And if you truly deliver on your promise over lunch, they might even offer to pick up the check.