As someone who is also hoping to complete a 2 year fellowship in neuro... I'd recommend not settling, as hard as it is. All of the postdocs I've looked at require an APA-accred internship, so I'm recommend (a) looking at some postdocs and seeing what they require, and (b) not going the APPIC route.
Regarding whether any random APA internship would help... I think it depends a lot on your training thus far. If you have a lot of neuro experience, you can sell it as, "I have a ton of neuro experience and I wanted to get a variety of experiences and perspectives to make myself a well-rounded psychologist," and it won't really count against you much. If you don't have a lot of neuro experience already, then you definitely want to pick some up during internship in order to make yourself a competitive applicant. Of course, this is just what I hear, since I'm clearly in the same boat as you. 🙂.
As much as it sucks, I'm not willing to go to a program that does not match my interests... even if it means taking another year.