This isn't the expected behavior - the day selection should work exactly the same way in both Lite and Pro versions.
Are you able to reproduce it in the online demo?
https://javascript.daypilot.org/demo/month/navigator.html
Also, the Chrome Developer Tools inspection features are not intentionally blocked. However, the right click handler prevents the default action (which is to open the browser context menu).
If you can only reproduce the problem locally, would you be able to provide the steps (and setup) to reproduce it? Then it should be no problem to fix it.