You’re not overriding – if you were, you’d use the keyword override. Instead, you’re using the keyword new, which hides the base class property in the derived class. Of course, you can’t override the property here, given that the type of ListView.View is just View, not CustView… and the property isn’t even virtual in the first place.
It’s not clear that you should be overriding anything though. Any code using just ListView will expect to be able to use View.LargeIcon etc. You’ve effectively got two properties here – if that’s what you want, you should give them different names to make it absolutely clear that you do have two properties. But you should also consider how you expect them to interact.
You haven’t told us what you’re really trying to achieve here – but it feels like your current approach is going to cause more problems than solutions. If the point is really just to avoid seeing options you don’t want in a drop-down, I’d probably add some sort of validation instead. (I don’t know which event it would be best to attach that to, admittedly. It’s been a while since I’ve done WinForms.)
(As an aside, I’d also recommend using automatically implemented properties rather than writing 12 lines of code where 1 line would do exactly the same thing.)