Install Tweetbot 3 on iOS 7
Does the store not give you an option to install the latest compatible version?

According to that article, you should be able to look into your list of purchased apps, and the App Store will ask you if you want to install the latest compatible version if your version of iOS does not support the current version of the app.

It's possible that this will only work for apps that you have previously purchased, but I don't have a device here to test.