Hey John,
Will LimeLM with TurboActivate meet these requirements, or can it be configured to do this?
Yep, you can use LimeLM & TurboActivate to do this. What you'll want to do is use custom license fields. For example, create a custom license field called "expires_on", set the date to 30 days from the creation date. Then, in your app read that "expires_on" value and use the IsDateValid() function to see if the customer can still use your app.
If the customer buys more days from you then you can update that value and they'll be able to use your app for X more days. Does that make sense?
Here's the best place to start: Getting started with LimeLM.