Still no time? Try this:
"I realize that you are a busy person and your time is limited. Yet we all take time for lunch. I would like to make you an offer. I would be happy to treat you to lunch at the restaurant or cafe of your choice, if you will just take the time to listen to my background while you're eating. All I ask of you is that you keep your ears open. Fair enough?"
If lunch doesn't work, offer breakfast. When your offer is accepted, either order light or not at all—use your time to sell the manager on you.
It will be the "cheapest" meal you will ever buy since your new contact will feel a personal obligation to assist you, either directly or indirectly.