Adding REST to Django
I ended up going with my own REST API framework for Django (that I'd love to get rid of if I can find a workable alternative), with a few custom views thrown in for corner cases I didn't want to deal with. It's worked out ok. 
So a combination of 1 and 2; without some form of framework you'll end up writing the same boilerplate for the common cases.
I've also done a few stand-alone APIs. I like having them as stand-alone services, but the very fact that they stand alone from the rest of the code leads to them getting neglected. No technical reason; simply out-of-sight, out-of-mind.
What I'd really like to see is an approach that unifies Django forms and REST APIs, as they often share a lot of logic. Conceptually if your app exposes something in HTML it likely wants to expose it programmatically as well.
I've got a Django application that works nicely.  I'm adding REST services.  I'm looking for some additional input on my REST strategy.  
Here are some examples of things I'm wringing my hands over.

Right now, I'm using the Django-REST API with a pile of patches.  
I'm thinking of falling back to simply writing view functions in Django that return JSON results.
I can also see filtering the REST requests in Apache and routing them to a separate, non-Django server instance.

Please nominate one approach per answer so we can vote them up or down.